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0 (a) Vr 4 3 V3 (a) ht t2 2t h2 (a) f y 3 y (a) f x x 8 2 (a) f 8 f 4 (b) (b) (b) (b) (b) (b) f 3 (cc) gs 2 (c) V2r 41. h1.5 (c) hx 2 f 0.25 (c) f 4x2 f 1 (c gx ht 1 2t 3 5 t ht 4 3 2 1 42 333202_0104.qxd 12/7/05 8:35 AM Page 50 50 43. Chapter 1 Functions and Their Graphs f x 1 2x 4, x โค 0 x 22 44. f x 9 x 2, x < 3 x 3 In Exercises 45โ52, find all real values of f x 0. x such that 45. 47. 49. 51. f x 15 3x f x 3x 4 f x x 2 9 f x x3 x 5 46. 48. 50. 52. f x 5x 1 f x 12 x2 5 f x x2 8x 15 f x x3 x2 4x 4 In Exercises 53โ56, find the value(s) of f x gx. x for which 53. 54. 55. 56. gx 3x 3 f x x2 2x 1, f x x4 2x 2, f x 3x 1, f x x 4, gx 2x2 gx x 1 gx 2 x In Exercises 57โ70, find the domain of the function. 57. 59. 61. 63. 65. 67. 69. f x 5x2 2x 1 ht 4 t g y y 10 f x 41 x2 gx 58. 60. 62. 64. 66. 68. 70. gx 1 2x2 s y 3y y 5 f t 3t 4 f x 4x2 3x hx 10 f x x 2 2x x 6 6 x f x x 5 x2 9 In Exercises 71โ74, assume that the domain of A {2, 1, 0, 1, 2}. the set pairs that represents the function is Determine the set of ordered f 71. f x x2 72. f. f x x2 3 73. f x x 2 74. f x x 1 Exploration |
In Exercises 75โ78, match the data with one of the following functions f x cx, gx cx2, hx cx, and rx c x and determine the value of the constant the function fit the data in the table. c that will make 75. 76. 77. 78 32 1 2 0 0 1 4 2 32 32 1 1 4 0 4 1 1 Undef. 32 In Exercises 79โ86, find the difference quotient and simplify your answer. 79. f x x 2 x 1 80. f x 5x x 2, 81. f x x3 3x, 82. f x 4x2 2x 83. 84. gx 1 x2, f t 1 t 2, 85. f x 5x,, x 3 gx g3 86. f x x23 1, x 8, 87. Geometry Write the area A of a square as a function of its perimeter P. A 88. Geometry Write the area of a circle as a function of its circumference C. The symbol indicates an example or exercise that highlights algebraic techniques specifically used in calculus. 333202_0104.qxd 12/7/05 8:35 AM Page 51 89. Maximum Volume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). x 24 2โ x x 24 2โ x x V (a) The table shows the volume x the box for various heights table to estimate the maximum volume. (in cubic centimeters) of (in centimeters). Use the Height, x 1 2 3 4 5 6 Volume, V 484 800 972 1024 980 864 (b) Plot the points x, V the relation defined by the ordered pairs represent a function of x? from the table in part (a). Does as V (c) If V is a function of write the function and determine x, its domain. 90. Maximum Profit The cost per unit in the production of a portable CD player is $60. The manufacturer charges $90 per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by $0.15 per CD player for each unit ordered in excess of 100 (for example, there would be a charge of $87 per CD player for an order size of 120). (a) The table shows the profit (in dollars) for various x. numbers of units |
ordered, Use the table to estimate the maximum profit. P Units, x 110 120 130 140 Profit, P 3135 3240 3315 3360 Units, x 150 160 170 Profit, P 3375 3360 3315 x, P (b) Plot the points from the table in part (a). Does the relation defined by the ordered pairs represent as a function of x? P (c) If P is a function of write the function and determine x, its domain. Section 1.4 Functions 51 91. Geometry A right triangle is formed in the first quadrant (see x, by the - and -axes and a line through the point A figure). Write the area of the triangle as a function of and determine the domain of the function. 20, )b (2, 1) a (, 0 = 36 โ x 2 (x, y) x โ6 โ4 โ2 2 4 6 x FIGURE FOR 91 FIGURE FOR 92 92. Geometry A rectangle is bounded by the -axis and the (see figure). Write the area of and determine the domain semicircle the rectangle as a function of of the function. y 36 x2 x, A x 93. Path of a Ball The height y (in feet) of a baseball thrown by a child is y 1 10 x2 3x 6 x where is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.) 94. Prescription Drugs The amounts (in billions of dollars) spent on prescription drugs in the United States from 1991 to 2002 (see figure) can be approximated by the model d dt 5.0t 37, 18.7t 64, 1 โค t โค 7 8 โค t โค 12 t represents the year, with where corresponding to 1991. Use this model to find the amount spent on prescription drugs in each year from 1991 to 2002. (Source: U.S. Centers for Medicare & Medicaid Services ( 180 150 120 90 60 30 1 2 3 4 5 6 7 8 9 10 11 12 Year (1 โ 1991) t 333202_0104.qxd 12/7/05 8:35 AM Page 52 52 Chapter 1 Functions and Their Graphs 95. Average Price The average prices (in thousands of dollars) of a new mobile home in the |
United States from 1990 to 2002 (see figure) can be approximated by the model p pt 0.182t2 0.57t 27.3, 2.50t 21.3, 0 โค t โค 7 8 โค t โค 12 t represents the year, with where corresponding to 1990. Use this model to find the average price of a mobile (Source: U.S. home in each year from 1990 to 2002. Census Bureau) t 0 p 55 50 45 40 35 30 25 20 15 10 Year (0 โ 1990) t 10 11 12 96. Postal Regulations A rectangular package to be sent by the U.S. Postal Service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure). (b) Write the revenue R as a function of the number of units sold. (c) Write the profit as a function of the number of units ) P R C sold. (Note: P 98. Average Cost The inventor of a new game believes that the variable cost for producing the game is $0.95 per unit and the fixed costs are $6000. The inventor sells each game for $1.69. Let be the number of games sold. x (a) The total cost for a business is the sum of the variable as a cost and the fixed costs. Write the total cost function of the number of games sold. C (b) Write the average cost per unit C Cx as a function of x. 99. Transportation For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate 8 0.05n 80, where the rate is given in dollars and people. is the number of n โฅ 80 n (a) Write the revenue R for the bus company as a function of n. (b) Use the function in part (a) to complete the table. What can you conclude? 90 100 110 120 130 140 150 n Rn F 100. Physics The force is a dam of Fy 149.7610 y52, (in feet). (in tons) of water against the face estimated function y is the depth of the water where the by x x y (a) Write the volume V What is the domain of the function? of the package as a function of (b) Use a graphing utility to graph your function. Be sure to use an appropriate window setting. (c) What dimensions will maximize the volume of the package? Explain your answer |
. 97. Cost, Revenue, and Profit A company produces a product for which the variable cost is $12.30 per unit and the fixed costs are $98,000. The product sells for $17.98. Let be the number of units produced and sold. x (a) The total cost for a business is the sum of the variable as a cost and the fixed costs. Write the total cost function of the number of units produced. C (a) Complete the table. What can you conclude from the table? y F y 5 10 20 30 40 x. (b) Use the table to approximate the depth at which the force against the dam is 1,000,000 tons. (c) Find the depth at which the force against the dam is 1,000,000 tons algebraically. 101. Height of a Balloon A balloon carrying a transmitter ascends vertically from a point 3000 feet from the receiving station. (a) Draw a diagram that gives a visual representation of represent the height of the balloon represent the distance between the balloon the problem. Let d and let and the receiving station. h (b) Write the height of the balloon as a function of d. What is the domain of the function? 333202_0104.qxd 12/7/05 8:35 AM Page 53 Model It Synthesis Section 1.4 Functions 53 102. Wildlife The graph shows the numbers of threatened and endangered fish species in the world from 1996 represent the number of threatthrough 2003. Let t. ened and endangered fish species in the year (Source: U.S. Fish and Wildlife Service) f t f t 126 125 124 123 122 121 120 119 118 117 116 1996 1998 2000 2002 Year t (a) Find f 2003 f 1996 2003 1996 in the context of the problem. and interpret the result N (b) Find a linear model for the data algebraically. represent the number of threatened and correspond Let endangered fish species and let to 1996. x 6 (c) Use the model found in part (b) to complete the table. 6 7 8 9 10 11 12 13 x N (d) Compare your results from part (c) with the actual data. (e) Use a graphing utility to find a linear model for the data. Let correspond to 1996. How does the model you found in part (b) compare with the model given by the graphing utility? x 6 True or False? In Exercises 103 and 104, determine whether the statement is true |
or false. Justify your answer. 103. The domain of the function given by f x x 4 1 is,, and the range of 104. The set of ordered pairs 0, 4, 2, 2 2, 2, 0,. f x is 8, 2, represents a function. 6, 0, 4, 0, 105. Writing In your own words, explain the meanings of domain and range. 106. Think About gx 3x 2. It Consider and Why are the domains of and different? f x x 2 g f In Exercises 107 and 108, determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. 107. (a) The sales tax on a purchased item is a function of the selling price. (b) Your score on the next algebra exam is a function of the number of hours you study the night before the exam. 108. (a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped. Skills Review In Exercises 109โ112, solve the equation. 109. 110. 111. 112 xx 1 12 In Exercises 113โ116, find the equation of the line passing through the pair of points. 113. 115. 2, 5, 6, 5, 3, 5 4, 1 114. 116. 10, 0, 1 2, 3, 1, 9 11 2, 1 3 333202_0105.qxd 12/7/05 8:36 AM Page 54 54 Chapter 1 Functions and Their Graphs 1.5 Analyzing Graphs of Functions What you should learn โข Use the Vertical Line Test for functions. โข Find the zeros of functions. โข Determine intervals on which functions are increasing or decreasing and determine relative maximum and relative minimum values of functions. โข Determine the average rate of change of a function. โข Identify even and odd functions. Why you should learn it Graphs of functions can help you visualize relationships between variables in real life. For instance, in Exercise 86 on page 64, you will use the graph of a function to represent visually the temperature for a city over a 24-hour period. y 5 4 1 (โ1, 1) Range โ3 โ2 (0, 3) y f x= ( ) (5, 2) 2 3 4 6 x (2, โ3) Domain โ |
5 FIGURE 1.53 The Graph of a Function In Section 1.4, you studied functions from an algebraic point of view. In this section, you will study functions from a graphical perspective. f x, f x such is in the domain of As you study this section, remember that The graph of a function is the collection of ordered pairs x x y f x the directed distance from the -axis the directed distance from the -axis f. x y that as shown in Figure 1.52. y 2 1 y = f(x) f(x) โ1 1 2 x x โ1 FIGURE 1.52 Example 1 Finding the Domain and Range of a Function Use the graph of the function f, (b) the function values f, f1 shown in Figure 1.53, to find (a) the domain of and, and (c) the range of f 2 f. Solution a. The closed dot at the open dot at is all of f b. Because 1, 1 indicates that x 5 5, 2 in the interval indicates that 1, 5. is a point on the graph of x 1, 1 x 1 Similarly, because f 2 3. 2, 3 is a point on the graph of f, f, it follows that f 1 1. it follows that is in the domain of whereas is not in the domain. So, the domain f, c. Because the graph does not extend below range of f is the interval 3, 3. f 2 3 or above f 0 3, the Now try Exercise 1. The use of dots (open or closed) at the extreme left and right points of a graph indicates that the graph does not extend beyond these points. If no such dots are shown, assume that the graph extends beyond these points. 333202_0105.qxd 12/7/05 8:36 AM Page 55 Section 1.5 Analyzing Graphs of Functions 55 By the definition of a function, at most one -value corresponds to a given x -value. This means that the graph of a function cannot have two or more different points with the same coordinate, and no two points on the graph of a function can be vertically above or below each other. It follows, then, that a vertical line can intersect the graph of a function at most once. This observation provides a convenient visual test called the Vertical Line Test for functions. x- y Vertical Line Test for Functions x A set of points in a coordinate plane is the graph of as a |
function of and only if no vertical line intersects the graph at more than one point. y if Example 2 Vertical Line Test for Functions Use the Vertical Line Test to decide whether the graphs in Figure 1.54 represent y as a function of x. y 4 2 x 1 โ3 โ2 โ1 (a) FIGURE 1.54 y 4 3 2 1 (b) y 4 3 1 โ1 โ1 (c Solution a. This is not a graph of as a function of because you can find a vertical line there is more that intersects the graph twice. That is, for a particular input than one output x, x, y. y b. This is a graph of as a function of y x, the graph at most once. That is, for a particular input output y. because every vertical line intersects there is at most one x, y c. This is a graph of as a function of x. (Note that if a vertical line does not intersect the graph, it simply means that the function is undefined for that particular value of there is at most one ) That is, for a particular input output x, x. y. Now try Exercise 9. 333202_0105.qxd 12/7/05 8:36 AM Page 56 56 Chapter 1 Functions and Their Graphs Zeros of a Function If the graph of a function of has an -intercept at function. x x a, 0, then a is a zero of the Zeros of a Function The zeros of a function of are the -values for which f x 0. f x x โ x 10 3 โ1 1 2 โ ( 2, 0) ( ) 5 3, 0 โ2 โ4 โ6 โ8 x Example 3 Finding the Zeros of a Function Find the zeros of each function. a. f x 3x2 x 10 b. gx 10 x2 c. ht 2t 3 t 5 Solution To find the zeros of a function, set the function equal to zero and solve for the independent variable. Zeros of f: FIGURE 1.55 x 22 โ4 ) ( โ 10, 0 โ6 โ4 โ2 ( ) = 10 โ 2 g x x ( 10, 0 ) 2 4 6 x ยฑ 10 Zeros of g: FIGURE 1.56 y 2 2(, 03 ) 2 4 6 h t( ) = 2t โ 3 t + 5 โ4 โ2 โ2 โ4 โ6 โ8 |
Zero of h: FIGURE 1.57 t 3 2 x t 3x2 x 10 0 3x 5x 2 0 3x 5 0 x 2 0 Set f x equal to 0. Factor. x 5 3 x 2 Set 1st factor equal to 0. Set 2nd factor equal to 0. x 5 3 as its -intercepts. x 2. and x 2, 0 In Figure 1.55, note that the graph of f a. b. f and The zeros of are 3, 0 5 has 10 x2 0 10 x2 0 10 x2 ยฑ 10 x Set gx equal to 0. Square each side. Add x2 to each side. Extract square roots. g The zeros of graph of has g c. 0 2t 3 t 5 2t 3 0 2t 3 t 3 2 are 10, 0 x 10 and and 10, 0 x 10. In Figure 1.56, note that the x as its -intercepts. Set ht equal to 0. Multiply each side by t 5. Add 3 to each side. Divide each side by 2. h The zero of its -intercept. t is t 3 2. In Figure 1.57, note that the graph of has h 2, 0 3 as Now try Exercise 15. 333202_0105.qxd 12/7/05 8:36 AM Page 57 Section 1.5 Analyzing Graphs of Functions 57 Increasing and Decreasing Functions 1 โ2 โ1 FIGURE 1.58 Increasing Constant 1 2 3 4 x y 1 f(x) = x3 The more you know about the graph of a function, the more you know about the function itself. Consider the graph shown in Figure 1.58. As you move from left x 2, to right, this graph falls from to and rises from is constant from x 2 x 0, x 4. x 2 x 0 to to Increasing, Decreasing, and Constant Functions x2 is increasing on an interval if, for any A function x1 < x2 f implies < f x2 f x1 and x1. in the interval, A function x1 < x2 f implies f x1 > f x2. is decreasing on an interval if, for any x1 and x2 in the interval, f A function f x 2 f x1 is constant on an interval if, for any. x1 and x2 in the interval, Example 4 Increasing and Decreasing Functions Use the |
graphs in Figure 1.59 to describe the increasing or decreasing behavior of each function. Solution a. This function is increasing over the entire real line. b. This function is increasing on the interval interval 1, 1, and increasing on the interval, 1, 1,. decreasing on the c. This function is increasing on the interval, 0, constant on the interval 0, 2, 2,. and decreasing on the interval y f(x) = x โ 3x 3 (โ1, 2) 2 y 2 1 (0, 1) (2, 1) โ1 x 1 โ2 โ1 1 2 x 1 2 3 t โ1 (a) FIGURE 1.59 โ1 โ2 (1, โ2) โ1 โ2 f(t) = t + 1, t < 0 1, 0 โค t โค 2 โt + 3, t > 2 (b) (c) Now try Exercise 33. To help you decide whether a function is increasing, decreasing, or constant on an interval, you can evaluate the function for several values of However, calculus is needed to determine, for certain, all intervals on which a function is increasing, decreasing, or constant. x. 333202_0105.qxd 12/7/05 8:36 AM Page 58 58 Chapter 1 Functions and Their Graphs A relative minimum or relative maximum is also referred to as a local minimum or local maximum. y Relative maxima Relative minima FIGURE 1.60 4 5 โ4 FIGURE 1.61 The points at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative minimum or relative maximum values of the function. Definitions of Relative Minimum and Relative Maximum A function value if there exists an interval is called a relative minimum of that contains fa f x1, x2 x1 < x < x2 a such that fa โค f x. implies fa is called a relative maximum of f if there exists an A function value interval x1, x2 x1 < x < x2 that contains implies a such that f a โฅ f x. Figure 1.60 shows several different examples of relative minima and relative maxima. In Section 2.1, you will study a technique for finding the exact point at which a second-degree polynomial function has a relative minimum or relative maximum. For the time being, however, you can use a graphing utility to find reasonable approximations of these points. x Example 5 Approximating a |
Relative Minimum Use a graphing utility to approximate the relative minimum of the function given by f x 3x2 4x 2. Solution is shown in Figure 1.61. By using the zoom and trace features or The graph of the minimum feature of a graphing utility, you can estimate that the function has a relative minimum at the point f 0.67, 3.33. Relative minimum Later, in Section 2.1, you will be able to determine that the exact point at which. the relative minimum occurs is 2 3, 10 Now try Exercise 49. 3 You can also use the table feature of a graphing utility to approximate numerically the relative minimum of the function in Example 5. Using a table that begins at 0.6 and increments the value of by 0.01, you can approximate that the minimum of occurs at the point 0.67, 3.33. f x 3x2 4x 2 x Te c h n o l o g y If you use a graphing utility to estimate the x- and y-values of a relative minimum or relative maximum, the zoom feature will often produce graphs that are nearly flat. To overcome this problem, you can manually change the vertical setting of the viewing window. The graph will stretch vertically if the values of Ymin and Ymax are closer together. 333202_0105.qxd 12/7/05 8:36 AM Page 59 Section 1.5 Analyzing Graphs of Functions 59 Average Rate of Change y (x1, f(x1)) (x2, f(x2)) Secant line f x2 โ x1 f(x2) โ f(x1) x1 x2 FIGURE 1.62 y f(x) = x 3 โ 3x 2 (0, 0) โ3 โ2 โ1 1 2 3 โ1 (โ2, โ2) (1, โ2) โ3 FIGURE 1.63 x x Exploration Use the information in Example 7 to find the average speed of 0 9 t1 the car from seconds. Explain why the result is less than the value obtained in part (b). to t2 In Section 1.3, you learned that the slope of a line can be interpreted as a rate of change. For a nonlinear graph whose slope changes at each point, the average rate of change between any two points is the slope of the line through the two points (see Figure 1.62). The line through the two points is called the |
secant line, and the slope of this line is denoted as x2, fx2 x1, fx1 and msec. Average rate of change of f from x1 to x2 fx2 fx1 x1 x2 change in y change in x msec Example 6 Average Rate of Change of a Function Find the average rates of change of 0 and (b) from 1 to x1 x2 (see Figure 1.63). fx x3 3x (a) from x1 2 to x2 0 from 0 is x2 f Solution a. The average rate of change of fx1 fx2 x1 x2 f b. The average rate of change of f1 f0 fx1 fx2 1 0 x1 x2 f0 f2 0 2 1. x1 2 to 0 2 2 0 x1 2 0 1 to x2 2. from 1 is Secant line has positive slope. Secant line has negative slope. Now try Exercise 63. Example 7 Finding Average Speed s The distance (in feet) a moving car is from a stoplight is given by the function st 20t32, t where is the time (in seconds). Find the average speed of the car 4 4 t2 (a) from to seconds and (b) from seconds. 9 0 to t2 t1 t1 Solution a. The average speed of the car from st1 st2 t1 t2 s4 s0 4 0 b. The average speed of the car from st1 st2 t1 t2 s9 s4 9 4 Now try Exercise 89. 0 to t1 t2 160 0 4 4 t1 t2 540 160 5 to 4 seconds is 40 9 feet per second. seconds is 76 feet per second. 333202_0105.qxd 12/7/05 8:36 AM Page 60 60 Chapter 1 Functions and Their Graphs Even and Odd Functions In Section 1.2, you studied different types of symmetry of a graph. In the terminology of functions, a function is said to be even if its graph is symmetric with respect to the -axis and to be odd if its graph is symmetric with respect to the origin. The symmetry tests in Section 1.2 yield the following tests for even and odd functions. y Exploration Graph each of the functions with a graphing utility. Determine whether the function is even, odd, or neither. f x x2 x 4 gx 2x 3 1 h |
x x 5 2x3 x jx 2 x6 x8 kx x 5 2x 4 x 2 px x9 3x5 x3 x What do you notice about the equations of functions that are odd? What do you notice about the equations of functions that are even? Can you describe a way to identify a function as odd or even by inspecting the equation? Can you describe a way to identify a function as neither odd nor even by inspecting the equation? Tests for Even and Odd Functions A function is even if, for each x in the domain of f, A function is odd if, for each x in the domain of f, y f x f x f x. y f x f x f x. Example 8 Even and Odd Functions a. The function gx x 3 x gx x3 x x3 x x 3 x gx is odd because gx gx, as follows. Substitute x for x. Simplify. Distributive Property Test for odd function b. The function hx x 2 1 is even because hx hx, as follows. hx x2 1 x 2 1 hx Substitute x for x. Simplify. Test for even function The graphs and symmetry of these two functions are shown in Figure 1.64. y 3 1 g(x) = x โ x 3 โ3 โ2 (โx, โy) โ1 โ2 โ3 (x, y) 1 2 3 x (โ x, y) y 6 5 4 3 2 (x, y) h(x) = x + 1 2 โ3 โ2 โ1 1 2 3 x (b) Symmetric to y-axis: Even Function (a) Symmetric to origin: Odd Function FIGURE 1.64 Now try Exercise 71. 333202_0105.qxd 12/7/05 8:36 AM Page 61 1.5 Exercises Section 1.5 Analyzing Graphs of Functions 61 VOCABULARY CHECK: Fill in the blanks. 1. The graph of a function f is the collection of ________ ________ or x, f x such that x is in the domain of f. 2. The ________ ________ ________ is used to determine whether the graph of an equation is a function of y in terms of x. 3. The ________ of a function are the values of f x 4. A function f is ________ on an interval if, for any for which x2 |
and x1 f x 0. in the interval, 5. A function value is a relative ________ of f if there exists an interval containing f a f a โฅ f x. implies implies x1 < x2 x1, x2 f x1 a. > f x2 x1 < x < x2 such that 6. The ________ ________ ________ ________ between any two points through the two points, and this line is called the ________ line. x1, f x1 and x2, f x2 is the slope of the line 7. A function 8. A function f f is ________ if for the each x in the domain of f, is ________ if its graph is symmetric with respect to the -axis. f x f x. y PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. y = f(x) โ4 โ2 In Exercises 1โ 4, use the graph of the function to find the domain and range of f. y = f(x) x 2 4 2. y 6 4 2 โ2 โ2 4. y = f(x) 1. 3. โ4 โ2 y 6 4 2 โ2 y 6 2 โ4 โ2 โ2 โ4 y = f(x) x 2 4 In Exercises 5โ8, use the graph of the function to find the indicated function values. (a) 6. (a) (b) (b) (d) (d) (c) (c) 2 y y = f(x) 4 3 2 โ3 โ4 y = f(x) y x 43 โ4 x 2 4 2 โ2 โ4 7. (a) (c) f 2 f 0 (b) (d) f 1 f 2 y = f(x) y 8. (a) (c) f 2 f 3 y (b) (d) f 1 f 1 y = f(x) x 2 4 โ2 โ4 โ6 4 2 โ2 โ2 x 2 4 In Exercises 9โ14, use the Vertical Line Test to determine whether y is a function of x.To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 9. y 1 2x 2 10. y 1 4x3 y 6 4 2 โ4 โ2 2 4 11 |
. x y 2 1 y 4 2 โ2 4 6 y 4 2 โ2 โ4 โ4 12. x2 y 2 25 x 2 4 y 6 4 2 โ 333202_0105.qxd 12/7/05 8:36 AM Page 62 62 Chapter 1 Functions and Their Graphs 13. x2 2xy 1 y 4 2 โ2 โ4 โ4 x 2 4 142 โ4 โ6 In Exercises 15โ24, find the zeros of the function algebraically. 15. 17. 19. 20. 21. 22. 23. 24. 16. 18. f x 3x2 22x 16 f x x2 9x 14 4x f x f x 2x2 7x 30 x 9x2 4 2 x3 x f x 1 f x x3 4x2 9x 36 f x 4x3 24x2 x 6 f x 9x4 25x2 f x 2x 1 f x 3x 2 In Exercises 25โ30, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. 25. 26. 27. 28. 29. 30. f x 3 5 x f x xx 7 f x 2x 11 f x 3x 14 8 f x 3x 1 x 6 f x 2x2 9 3 x In Exercises 31โ38, determine the intervals over which the function is increasing, decreasing, or constant. 31. f x 3 2 x y 4 2 32. f x x2 4x y โ4 โ2 2 4 x โ4 โ2 2 x 6 โ2 โ4 (2, โ4) 33. f x x3 3x2 2 34. f x x2 1 y 4 2 (0, 2) โ2 2 4 x (2, โ2) (โ1, 0) โ4 โ2 35. f x x 3, 3, 2x 12 (1, 0) 2 4 x y 6 4 โ2 2 4 x 36. f x 2x 1, x2 22 2 4 x โ4 37. f x x 1 x 1 38. y 6 4 (โ1, 2) (1, 2) โ2 2 4 x f x x2 x 1 x 1 y (0, 1) โ4 โ2 (โ2, โ3) โ2 x 2 333202_0105.qxd |
12/7/05 8:36 AM Page 63 In Exercises 39โ 48, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a). 39. 41. 43. 45. 47. f x 3 gs s2 32 40. gx x 42. hx x2 4 44. 46. 48. f x 3x4 6x2 f x xx 3 f x x23 In Exercises 49โ54, use a graphing utility to graph the function and approximate (to two decimal places) any relative minimum or relative maximum values. 49. 51. 53. 54. f x x 4x 2 f x x2 3x 2 f x xx 2x 3 f x x3 3x2 x 1 50. 52. f x 3x2 2x 5 f x 2x2 9x 55. f x โฅ 0. In Exercises 55โ 62, graph the function and determine the interval(s) for which 4x 2 f x x 2 4x 62. 58. 59. 56. 61. 57. 60. In Exercises 63โ70, find the average rate of change of the function from to x1 x2. Function 63. 64. 65. 66. 67. 68. 69. 70. f x 2x 15 f (x 3x 8 f x x2 12x 4 f x x2 2x 8 f x x3 3x2 x f x x3 6x2 -Values 0, x2 0, x2 1, x2 1, x2 1, x2 1, x2 3, x2 3, x2 3 3 5 5 3 6 11 8 x1 x1 x1 x1 x1 x1 x1 x1 In Exercises 71โ76, determine whether the function is even, odd, or neither. Then describe the symmetry. 71. 73. 75. f x x6 2x2 3 gx x3 5x f t t 2 2t 3 72. 74. 76. hx x3 5 f x x1 x 2 gs 4s 23 Section 1.5 Analyzing Graphs of Functions 63 In Exercises 77โ80, write the height of the rectangle as a function of |
x. h 77. y 781, 2) (3, 2) 4 3 2 1 792, 4) h y x= 2 x1 1, 3 80. y 4 h โ2 (8, 2) 2 x 8 4 y 6 x= 3 x x In Exercises 81โ 84, write the length of the rectangle as a function of y. L 81. y 82. y x 6 4 y โ2 L (8, 4) x = 21 = 23 y (2, 4) L 1 2 3 4 83. y 84. y 4 3 2 y y= 2 x (4, 21, 2 85. Electronics The number of lumens (time rate of flow of from a fluorescent lamp can be approximated by L light) the model L 0.294x2 97.744x 664.875, 20 โค x โค 90 where x is the wattage of the lamp. (a) Use a graphing utility to graph the function. (b) Use the graph from part (a) to estimate the wattage necessary to obtain 2000 lumens. 333202_0105.qxd 12/7/05 8:36 AM Page 64 64 Chapter 1 Functions and Their Graphs Model It 88. Geometry Corners of equal size are cut from a square with sides of length 8 meters (see figure). a) Write the area of the resulting figure as a function of A x. Determine the domain of the function. (b) Use a graphing utility to graph the area function over its domain. Use the graph to find the range of the function. (c) Identify the figure that would result if were chosen to be the maximum value in the domain of the function. What would be the length of each side of the figure? x 89. Digital Music Sales The estimated revenues (in billions of dollars) from sales of digital music from 2002 to 2007 can be approximated by the model r r 15.639t3 104.75t2 303.5t 301, 2 โค t โค 7 t where 2002. represents the year, with (Source: Fortune) t 2 corresponding to (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 2002 to 2007. Interpret your answer in the context of the problem. 90. Foreign College Students The numbers of foreign students (in thousands) enrolled in colleges in the United States from 1992 to 2002 can |
be approximated by the model. F F 0.004t 4 0.46t2 431.6, 2 โค t โค 12 t where 1992. represents the year, with (Source: Institute of International Education) corresponding to t 2 (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 1992 to 2002. Interpret your answer in the context of the problem. (c) Find the five-year time periods when the rate of change was the greatest and the least. 86. Data Analysis: Temperature The table shows the (in degrees Fahrenheit) of a certain city represent the time of day, y temperature over a 24-hour period. Let where corresponds to 6 A.M. x 0 x Time, x Temperature, y 0 2 4 6 8 10 12 14 16 18 20 22 24 34 50 60 64 63 59 53 46 40 36 34 37 45 A model that represents these data is given by y 0.026x3 1.03x2 10.2x 34, 0 โค x โค 24. (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use the graph to approximate the times when the temperature was increasing and decreasing. (d) Use the graph to approximate the maximum and minimum temperatures during this 24-hour period. (e) Could this model be used to predict the temperature for the city during the next 24-hour period? Why or why not? 87. Coordinate Axis Scale Each function models the specit 5 fied data for the years 1995 through 2005, with corresponding to 1995. Estimate a reasonable scale for the vertical axis (e.g., hundreds, thousands, millions, etc.) of the graph and justify your answer. (There are many correct answers.) f t f t f t that is unemployed. represents the average salary of college professors. represents the percent of the civilian work force represents the U.S. population. (b) (c) (a) 333202_0105.qxd 12/7/05 8:36 AM Page 65 Physics In Exercises 91โ 96, (a) use the position equation s 16t2 v0t s0 to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of |
the function (d) interpret your answer to part (c) in the from context of the problem, (e) find the equation of the secant line through and (f) graph the secant line in the same viewing window as your position function. and to t2, t2, t1 t1 91. An object is thrown upward from a height of 6 feet at a velocity of 64 feet per second. t1 0, t2 3 92. An object is thrown upward from a height of 6.5 feet at a velocity of 72 feet per second. t1 0, t2 4 93. An object is thrown upward from ground level at a veloc- ity of 120 feet per second. t1 3, t2 5 94. An object is thrown upward from ground level at a veloc- ity of 96 feet per second. t1 2, t2 5 Section 1.5 Analyzing Graphs of Functions 65 Think About It In Exercises 101โ104, find the coordinates of a second point on the graph of a function if the given point is on the graph and the function is (a) even and (b) odd. f 101. 102. 103. 104. 2, 4 3, 7 3 5 4, 9 5, 1 105. Writing Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) (c) (e) y x y x3 y x5 (b) (d) (f 106. Conjecture Use the results of Exercise 105 to make a Use a conjecture about the graphs of graphing utility to graph the functions and compare the results with your conjecture. y x8. y x 7 and Skills Review 95. An object is dropped from a height of 120 feet. In Exercises 107โ110, solve the equation. t1 0, t2 2 96. An object is dropped from a height of 80 feet. t1 1, t2 2 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 97 and 98, determine whether 97. A function with a square root cannot have a domain that is the set of real numbers. 98. It is possible for an odd function to have the interval 0, as its domain. 99. If f g is an even function, determine whether is even, odd, (a) (b) or neither. Explain. gx |
f x gx f x gx f x 2 gx f x 2 (d) (c) 100. Think About It Does the graph in Exercise 11 represent Explain. as a function of y? x 107. 108. 109. 110. x2 10x 0 100 x 52 0 x3 x 0 16x2 40x 25 0 In Exercises 111โ114, evaluate the function at each specified value of the independent variable and simplify. 111. 112. 113. 114. f 4 f x 5x 8 f 9 (a) (b) f x x2 10x (a) (b) f x x 12 9 (a) (b) f x x4 x 5 (a) (b) f 1 f 12 f 4 f 8 f 40 (c) f x 7 (c) f x 4 f 36 (c) f 1 2 f 23 (c) In Exercises 115 and 116, find the difference quotient and simplify your answer. 115. f x x2 2x 9, 116. f x 5 6x x2 333202_0106.qxd 12/7/05 8:40 AM Page 66 66 Chapter 1 Functions and Their Graphs 1.6 A Library of Parent Functions What you should learn โข Identify and graph linear and squaring functions. โข Identify and graph cubic, square root, and reciprocal functions. โข Identify and graph step and other piecewise-defined functions. โข Recognize graphs of parent functions. Why you should learn it Step functions can be used to model real-life situations. For instance, in Exercise 63 on page 72, you will use a step function to model the cost of sending an overnight package from Los Angeles to Miami. Linear and Squaring Functions One of the goals of this text is to enable you to recognize the basic shapes of the graphs of different types of functions. For instance, you know that the graph of the linear function intercept at 0, b. The graph of the linear function has the following characteristics. is a line with slope f x ax b m a and y- โข The domain of the function is the set of all real numbers. โข The range of the function is the set of all real numbers. โข The graph has an -intercept of bm, 0 y x and a -intercept of 0, b. โข The graph is increasing if if m 0. m > 0, decreasing if m < 0, and constant Example 1 Writing a Linear Function |
Write the linear function f for which f 1 3 and f 4 0. Solution To find the equation of the line that passes through x2, y2 m y2 x2 4, 0, y1 x1 first find the slope of the line. 0 3 3 4 1 3 1 x1, y1 1, 3 and Next, use the point-slope form of the equation of a line. mx x1 y y1 y 3 1x 1 y x 4 f x x 4 Point-slope form Substitute for x1, y1, and m. Simplify. Function notation The graph of this function is shown in Figure 1.65. ยฉ Getty Images y 5 4 3 2 1 f(x) = โ1 โ1 FIGURE 1.65 Now try Exercise 1. 333202_0106.qxd 12/7/05 8:40 AM Page 67 Section 1.6 A Library of Parent Functions 67 There are two special types of linear functions, the constant function and the identity function. A constant function has the form f x c and has the domain of all real numbers with a range consisting of a single real number The graph of a constant function is a horizontal line, as shown in Figure 1.66. The identity function has the form c. f x x. m 1 Its domain and range are the set of all real numbers. The identity function has a slope of The graph of the identity function is a line for which each -coordinate equals the corresponding -coordinate. The graph is always increasing, as shown in Figure 1.67 and a -intercept 0, 0. y x y y 3 2 1 f(x) = c 1 2 3 x y f(x) = x 2 1 โ2 โ1 1 2 x โ1 โ2 FIGURE 1.66 FIGURE 1.67 The graph of the squaring function f x x2 is a U-shaped curve with the following characteristics. โข The domain of the function is the set of all real numbers. โข The range of the function is the set of all nonnegative real numbers. โข The function is even. โข The graph has an intercept at โข The graph is decreasing on the interval and increasing on, 0 0, 0. the interval 0,. โข The graph is symmetric with respect to the -axis. โข The graph has a relative minimum at 0, 0. y The graph of the squaring function is shown in Figure 1.68. y f |
(x3 โ2 โ1 โ1 2 1 (0, 0) 3 x FIGURE 1.68 333202_0106.qxd 12/7/05 8:40 AM Page 68 68 Chapter 1 Functions and Their Graphs Cubic, Square Root, and Reciprocal Functions The basic characteristics of the graphs of the cubic, square root, and reciprocal functions are summarized below. 1. The graph of the cubic function f x x3 has the following characteristics. โข The domain of the function is the set of all real numbers. โข The range of the function is the set of all real numbers. โข The function is odd. โข The graph has an intercept at 0, 0. โข The graph is increasing on the interval,. โข The graph is symmetric with respect to the origin. The graph of the cubic function is shown in Figure 1.69. f x x 2. The graph of the square root function has the following characteristics. โข The domain of the function is the set of all nonnegative real numbers. โข The range of the function is the set of all nonnegative real numbers. 0, 0. โข The graph has an intercept at โข The graph is increasing on the interval 0,. The graph of the square root function is shown in Figure 1.70. 3. The graph of the reciprocal function characteristics. f x 1 x has the following โข The domain of the function is, 0 0,. โข The range of the function is, 0 0,. โข The function is odd. โข The graph does not have any intercepts. โข The graph is decreasing on the intervals, 0 โข The graph is symmetric with respect to the origin. and 0,. The graph of the reciprocal function is shown in Figure 1.71. y 3 2 1 (0, 0) โ1 โ2 โ3 โ3 โ2 f(x0, 0) โ1 โ1 โ2 f(x(x) = 1 x โ1 1 2 3 x Cubic function FIGURE 1.69 Square root function FIGURE 1.70 Reciprocal function FIGURE 1.71 333202_0106.qxd 12/7/05 8:40 AM Page 69 Section 1.6 A Library of Parent Functions 69 Step and Piecewise-Defined Functions Functions whose graphs resemble sets of stairsteps are known as step functions. The most famous of the step functions is the greatest integer function, which is denoted by and defined as x f |
x x the greatest integer less than or equal to x. Some values of the greatest integer function are as follows. 1 greatest integer โค 1 1 1 2 greatest integer โค 1 1 10 greatest integer โค 1 1 1.5 greatest integer โค 1.5 1 0 10 2 The graph of the greatest integer function f x x y 3 2 1 โ4 โ3 โ2 โ1 1 2 3 4 x ( ) = [[ ]] f x x โ3 โ4 FIGURE 1.72 has the following characteristics, as shown in Figure 1.72. Te c h n o l o g y When graphing a step function, you should set your graphing utility to dot mode. โข The domain of the function is the set of all real numbers. โข The range of the function is the set of all integers. โข The graph has a -intercept at y 0, 0 and -intercepts in the interval x 0, 1. โข The graph is constant between each pair of consecutive integers. โข The graph jumps vertically one unit at each integer value. Example 2 Evaluating a Step Function Evaluate the function when x 1, 2, and 3 22 FIGURE 1.73 For f x ( ) = [[ ]] x + 1 โ3 โ2 โ1 1 2 3 4 5 x For Solution For 1, so โค 1 x 1, the greatest integer is f 1 1 1 1 1 0. x 2, โค 2 the greatest integer f 2 2 1 2 1 3. x 3 โค 3 2, the greatest integer 2 3 2 1 1 1 2. f 3 is 2, so is so 1, 2 You can verify your answers by examining the graph of Figure 1.73. f x x 1 shown in Now try Exercise 29. Recall from Section 1.4 that a piecewise-defined function is defined by two or more equations over a specified domain. To graph a piecewise-defined function, graph each equation separately over the specified domain, as shown in Example 3. 333202_0106.qxd 12/7/05 8:40 AM Page 70 70 Chapter 1 Functions and Their Graphs y โ x= + 4 21 = + 3 โ5 โ4 โ3 โ1 โ2 โ3 โ 4 โ5 โ6 FIGURE 1.74 Example 3 Graphing a Piecewise-Defined Function Sketch the graph of f x 2x 3, x 4, x โค 1 x > 1. Solution This piecewise-defined function is composed of two |
linear functions. At and to the left of and to the right of the graph is the line 1, 5 f1 21 3 5. x 1 x 1 as shown in Figure 1.74. Notice that the point is an open dot. This is because is a solid dot and the point the graph is the line y x 4, y 2x 3, x 1 1, 3 Now try Exercise 43. Parent Functions The eight graphs shown in Figure 1.75 represent the most commonly used functions in algebra. Familiarity with the basic characteristics of these simple graphs will help you analyze the shapes of more complicated graphsโin particular, graphs obtained from these graphs by the rigid and nonrigid transformations studied in the next section. y 3 2 1 f(x1 โ2 โ2 โ1 y f(x) = x y f(x) = ๏ฃฌx ๏ฃฌ x 1 2 โ2 โ1 2 1 โ1 โ2 x 1 2 y 3 2 1 f(x) = x 1 2 3 (a) Constant Function (b) Identity Function (c) Absolute Value Function (d) Square Root Function y 4 3 2 1 โ2 โ1 y 2 1 โ1 โ2 x 1 2 f(x) = x3 y 3 2 1 f(x3 โ2 โ1 1 2 3 ( ) = [[ ]] f x x โ3 โ2 โ1 f(x) = x2 x 1 2 x x (e) Quadratic Function FIGURE 1.75 (f) Cubic Function (g) Reciprocal Function (h) Greatest Integer Function 333202_0106.qxd 12/7/05 8:40 AM Page 71 1.6 Exercises VOCABULARY CHECK: Match each function with its name. 1. 4. 7. f x x f x x2 f x x 2. 5. 8. f x x f x x f x x3 (a) squaring function (b) square root function Section 1.6 A Library of Parent Functions 71 3. 6 ax b (c) cubic function 9. (d) linear function (e) constant function (f) absolute value function (e) greatest integer function (h) reciprocal function (i) identity function PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. f 1. 3. f 0 6 In Exercises 1 |
โ8, (a) write the linear function such that it has the indicated function values and (b) sketch the graph of the function. f 1 4, f 5 4, f 5 1, f 10 12, f 1 6, 15 f 2 2, f 2 17 f 5 1 f 16 1 f 4 3 f 4 11 f 1 2 f 1 11 f 3 8, f 3 9, 4. 2. 8. 6. 7. 5. 2 3 In Exercises 9โ28, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27 x2 2x hx x2 4x 12 f x x3 1 f x x 13 2 f x 4x gx 2 x 4 f x 1 x hx 1 x 2 10. 12. 14. 16. 18. 20. 22. 24. 26. 28. f x 3x 5 2 f x 5 2 3x 6 f x x2 8x gx x2 6x 16 f x 8 x3 gx 2x 33 1 f x 4 2x hx x 2 3 f x 4 1 x kx 1 x 3 In Exercises 29โ36, evaluate the function for the indicated values. 29. 30. f x x f 2.1 (a) gx 2x g3 (a) (b) f 2.9 (c) f 3.1 (d) f 7 2 (b) g0.25 (c) g9.5 (d) g 11 3 31. 32. 33. 34. 35. 36. 2 h1 f 0 hx x 3 h2 (a) (b) f x 4x 7 (a) (b) hx 3x 1 h2.5 (a) (b) kx 1 2x 6 k5 (a) (b) gx 3x 2 5 (a) gx 7x 4 6 g9 (a) g 2.7 g1 (b) (b) 8 f 1.5 h3.2 k6.1 g 1 (c) h4.2 (d) h21.6 (c) f 6 (d) f 5 3 (c) h7 3 (d) h21 3 (c) k0.1 (d) k15 (c) g 0.8 ( |
d) g 14.5 (c) g4 (d) g3 2 In Exercises 37โ42, sketch the graph of the function. 37. 39. 41. gx x gx x 2 gx x 1 38. 40. 42. gx 4 x gx x 1 gx x 3 In Exercises 43โ50, graph the function. 43. 44. 45. 46. 47 2x 3, 3 x, gx x 6, 2x 4, f x 4 x, 4 x, f x 1 x 12, x 2, f x x 2 5, x 2 4x 3 333202_0106.qxd 12/7/05 8:40 AM Page 72 72 48. 49. 50. Chapter 1 Functions and Their Graphs hx 3 x2, x2 2, hx 4 x2, kx 2x 1, 3 x, x2 1, 2x2 1, 1 x2 In Exercises 51 and 52, (a) use a graphing utility to graph the function, (b) state the domain and range of the function, and (c) describe the pattern of the graph. gx 21 sx 21 51. 52. 4x 1 4x 4x 1 4x2 In Exercises 53โ60, (a) identify the parent function and the transformed parent function shown in the graph, (b) write an equation for the function shown in the graph, and (c) use a graphing utility to verify your answers in parts (a) and (b). 53. y 54. y 4 2 โ2 โ4 2 1 543 โ6 โ4 55. y 2 1 โ1 โ2 โ3 โ4 57. 592 โ1 321 โ2 โ1 32 โ2 โ2 โ1 321 56. y 1 โ2 โ1 32 โ2 58. 60. y 2 1 โ4 โ2 โ1 1 โ3 โ4 y 2 1 โ2 โ1 32 โ4 61. Communications The cost of a telephone call between Denver and Boise is $0.60 for the first minute and $0.42 for each additional minute or portion of a minute. A model for the total cost (in dollars) of the phone call is C 0.60 0.421 t, where is the length of the phone call in minutes. t > 0 C t (a) Sketch the graph of the model. (b) Determine |
the cost of a call lasting 12 minutes and 30 seconds. 62. Communications The cost of using a telephone calling card is $1.05 for the first minute and $0.38 for each additional minute or portion of a minute. (a) A customer needs a model for the cost of using a calling card for a call lasting minutes. Which of the following is the appropriate model? Explain. C1 C2 t 1.05 0.38t 1 t 1.05 0.38t 1 C t (b) Graph the appropriate model. Determine the cost of a call lasting 18 minutes and 45 seconds. 63. Delivery Charges The cost of sending an overnight package from Los Angeles to Miami is $10.75 for a package weighing up to but not including 1 pound and $3.95 for each additional pound or portion of a pound. A model for the total cost (in dollars) of sending the package is C 10.75 3.95x, x is the weight in pounds. where x > 0 C (a) Sketch a graph of the model. (b) Determine the cost of sending a package that weighs 10.33 pounds. 64. Delivery Charges The cost of sending an overnight package from New York to Atlanta is $9.80 for a package weighing up to but not including 1 pound and $2.50 for each additional pound or portion of a pound. (a) Use the greatest integer function to create a model for the cost of overnight delivery of a package weighing x C pounds, x > 0. (b) Sketch the graph of the function. 65. Wages A mechanic is paid $12.00 per hour for regular time and time-and-a-half for overtime. The weekly wage function is given by Wh 12h, 18h 40 480, 0 < h โค 40 h > 40 where h is the number of hours worked in a week. (a) Evaluate W30, W40, W45, and W50. (b) The company increased the regular work week to 45 hours. What is the new weekly wage function? 333202_0106.qxd 12/7/05 8:40 AM Page 73 Section 1.6 A Library of Parent Functions 73 66. Snowstorm During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of 0.5 inch per hour |
for the final hour. Write and graph a piecewise-defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm? Model It V 100 75 50 25 ) 60, 100) (10, 75) (20, 75) (45, 50) (5, 50) (50, 50) (30, 25) (40, 25) 20 30 50 Time (in minutes) 40 t 60 (0, 0) 10 FIGURE FOR 68 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 69 and 70, determine whether 69. A piecewise-defined function will always have at least one y x -intercept or at least one -intercept, 4, 6, 70. can be rewritten as f x 2x, 1 โค x < 4. Exploration In Exercises 71 and 72, write equations for the piecewise-defined function shown in the graph. 71. y 72. y 8 6 4 2 (0, 6) (3, 2) (8, 0) x 2 4 6 8 10 8 6 4 (3, 4) (7, 0) (1, 1) (โ 1, 1) 4 (0, 0) 6 x Skills Review 67. Revenue The table shows the monthly revenue (in thousands of dollars) of a landscaping business for each month of the year 2005, with representing January. x 1 y Month, x Revenue 10 11 12 5.2 5.6 6.6 8.3 11.5 15.8 12.8 10.1 8.6 6.9 4.5 2.7 A mathematical model that represents these data is f x 1.97x 26.3. 0.505x2 1.47x 6.3 (a) What is the domain of each part of the piecewisedefined function? How can you tell? Explain your reasoning. (b) Sketch a graph of the model. (c) Find and the context of the problem. f 11, f 5 and interpret your results in (d) How do the values obtained from the model in part (b) compare with the actual data values? In Exercises 73 and 74, solve the inequality and sketch the solution on the real number line. 73. 3x 4 โค 12 5x 74. 2x 1 > 6x 9 68. Fluid Flow The intake pipe of a 100-gallon tank has |
a flow rate of 10 gallons per minute, and two drainpipes have flow rates of 5 gallons per minute each. The figure shows t. the volume Determine the combination of the input pipe and drain pipes in which the fluid is flowing in specific subintervals of the 1 hour of time shown on the graph. (There are many correct answers.) of fluid in the tank as a function of time V and passing through the pairs of points are parallel, perpen- In Exercises 75 and 76, determine whether the lines L2 dicular, or neither. L1 75. L1: 2, 2, 2, 10 L2: 1, 3, 3, 9 76. L1: 1, 7, 4, 3 L2: 1, 5, 2, 7 333202_0107.qxd 12/7/05 8:41 AM Page 74 74 Chapter 1 Functions and Their Graphs 1.7 Transformations of Functions What you should learn โข Use vertical and horizontal shifts to sketch graphs of functions. โข Use reflections to sketch graphs of functions. โข Use nonrigid transformations to sketch graphs of functions. Why you should learn it Knowing the graphs of common functions and knowing how to shift, reflect, and stretch graphs of functions can help you sketch a wide variety of simple functions by hand. This skill is useful in sketching graphs of functions that model real-life data, such as in Exercise 68 on page 83, where you are asked to sketch the graph of a function that models the amounts of mortgage debt outstanding from 1990 through 2002. Shifting Graphs Many functions have graphs that are simple transformations of the parent graphs summarized in Section 1.6. For example, you can obtain the graph of hx x2 2 by shifting the graph of h function notation, hx x2 2 f x x2 f and are related as follows. f x 2 Upward shift of two units upward two units, as shown in Figure 1.76. In Similarly, you can obtain the graph of gx x 22 f x x 2 by shifting the graph of f g In this case, the functions and have the following relationship. f x 2 gx x 22 Right shift of two units to the right two units, as shown in Figure 1.77. h(x) = x2 + 2 y 4 3 1 f(x) = x2 f(x) = x2 y g(x) = (x โ 2)2 4 3 2 1 โ2 โ1 |
1 2 x โ1 1 2 3 x FIGURE 1.76 FIGURE 1.77 ยฉ Ken Fisher/Getty Images The following list summarizes this discussion about horizontal and vertical shifts. Vertical and Horizontal Shifts Let be a positive real number. Vertical and horizontal shifts in the graph of are represented as follows. c y f x In items 3 and 4, be sure you hx f x c see that corresponds to a right shift and hx f x c corresponds to a left shift for c > 0. 1. Vertical shift units upward: c 2. Vertical shift units downward: c 3. Horizontal shift units to the right: c 4. Horizontal shift units to the left: c hx f x c hx f x c hx f x c hx f x c 333202_0107.qxd 12/7/05 8:41 AM Page 75 Section 1.7 Transformations of Functions 75 Some graphs can be obtained from combinations of vertical and horizontal shifts, as demonstrated in Example 1(b). Vertical and horizontal shifts generate a family of functions, each with the same shape but at different locations in the plane. Example 1 Shifts in the Graphs of a Function f x x3 Use the graph of gx x3 1 hx x 23 1 b. a. to sketch the graph of each function. Solution a. Relative to the graph of f x x3, the graph of gx x3 1 is a downward shift of one unit, as shown in Figure 1.78. b. Relative to the graph of involves a left shift of two units and an upward shift of one unit, as shown in Figure 1.79. the graph of hx x 23 1 f x x3(x) = (x + 2) + 1 3 y f(x2 โ1 1 2 x โ4 โ2 โโ3 x โ2 FIGURE 1.78 FIGURE 1.79 Now try Exercise 1. โ1 โ2 โ3 In Figure 1.79, notice that the same result is obtained if the vertical shift precedes the horizontal shift or if the horizontal shift precedes the vertical shift. Exploration Graphing utilities are ideal tools for exploring translations of functions. Graph try to predict how the graphs of and in same viewing window. Before looking at the graphs, relate to the graph of and f, g, f. h h g a. f x x 2, gx x 42, hx x 42 3 f x x 2, |
gx x 12, hx x 12 2 b. c. f x x 2, gx x 42, hx x 42 2 333202_0107.qxd 12/7/05 8:41 AM Page 76 76 Chapter 1 Functions and Their Graphs y 2 1 โ1 โ2 โ2 โ1 FIGURE 1.80 Reflecting Graphs The second common type of transformation is a reflection. For instance, if you consider the -axis to be a mirror, the graph of x f (x) = x2 1 2 h(x) = โx 2 hx x2 is the mirror image (or reflection) of the graph of x f x x2, as shown in Figure 1.80. Reflections in the Coordinate Axes Reflections in the coordinate axes of the graph of as follows. 1. Reflection in the -axis: x 2. Reflection in the -axis: y hx f x hx f x y f x are represented ( ) = 4 f x x 3 Example 2 Finding Equations from Graphs โ3 3 The graph of the function given by f x x 4 โ1 FIGURE 1.81 is shown in Figure 1.81. Each of the graphs in Figure 1.82 is a transformation of the graph of Find an equation for each of these functions. f. 1 โ3 โ1 (b) 5 y h x= ( ) 3 โ3 3 y g x= ( ) โ1 (a) FIGURE 1.82 Solution a. The graph of g units of the graph of gx x 4 2. Exploration Reverse the order of transformations in Example 2(a). Do you obtain the same graph? Do the same for Example 2(b). Do you obtain the same graph? Explain. is a reflection in the -axis followed by an upward shift of two x f x x 4. So, the equation for g is b. The graph of h reflection in the -axis of the graph of is a horizontal shift of three units to the right followed by a x So, the equation for f x x 4. is h hx x 34. Now try Exercise 9. 333202_0107.qxd 12/7/05 8:41 AM Page 77 Section 1.7 Transformations of Functions 77 Example 3 Reflections and Shifts Compare the graph of each function with the graph of f x x. a. gx x b. hx x c. kx x 2 Al |
gebraic Solution a. The graph of g is a reflection of the graph of f x in the -axis because gx x f x. b. The graph of h is a reflection of the graph of f y in the -axis because hx x f x. c. The graph of is a left shift of two units x followed by a reflection in the -axis because k kx x 2 f x 2. Graphical Solution f a. Graph and on the same set of coordinate axes. From the graph is a reflection of g g b. Graph and on the same set of coordinate axes. From the graph is a reflection of h in Figure 1.83, you can see that the graph of x in the -axis. the graph of f h f in Figure 1.84, you can see that the graph of y in the -axis. the graph of f f k c. Graph and on the same set of coordinate axes. From the graph is a left shift of two x k in Figure 1.85, you can see that the graph of f, units of the graph of followed by a reflection in the -axis. y 2 1 f(x) = x โ1 1 2 3 โ1 โ2 g(x) = xโ FIGURE 1.83 f(x) = x x 1 2 h(x) = โ x โ2 โ1 FIGURE 1.84 Now try Exercise 19. FIGURE 1.85 When sketching the graphs of functions involving square roots, remember that the domain must be restricted to exclude negative numbers inside the radical. For instance, here are the domains of the functions in Example 3. Domain of Domain of Domain of gx x: hx x: kx x 2 333202_0107.qxd 12/7/05 8:41 AM Page 78 78 Chapter 1 Functions and Their Graphs Nonrigid Transformations y h(x) = 3 ๏ฃฌx ๏ฃฌ 4 3 2 y 4 โ2 โ1 FIGURE 1.86 f(x) = ๏ฃฌx ๏ฃฌ x 1 2 g(x) = 1 3 ๏ฃฌx ๏ฃฌ f(x) = ๏ฃฌx ๏ฃฌ 2 1 y 6 โ 2 โ1 FIGURE 1.87 x 1 2 Horizontal shifts, vertical shifts, and reflections are rigid transformations because the basic shape of the graph is unchanged. These transformations change only the position of the graph in the coordinate plane. Nonrig |
id transformations are those that cause a distortionโa change in the shape of the original graph. For is represented instance, a nonrigid transformation of the graph of gx cf x, by and a vertical shrink if Another nonrigid transformation of the graph of y f x where the transformation is a horizontal shrink if where the transformation is a vertical stretch if is represented by c > 1 and a horizontal stretch if hx f cx, 0 < c < 1. 0 < c < 1. y f x c > 1 Example 4 Nonrigid Transformations Compare the graph of each function with the graph of gx 1 hx 3x b. a. 3x f x x. Solution a. Relative to the graph of hx 3x 3f x f x x, the graph of is a vertical stretch (each -value is multiplied by 3) of the graph of Figure 1.86.) y f. (See b. Similarly, the graph of gx 1 3x 3 f x 1 y is a vertical shrink each -value is multiplied by Figure 1.87.) 1 3 of the graph of f. (See g(x) = 2 โ 8x 3 Now try Exercise 23. Example 5 Nonrigid Transformations f(x) = 2 โ x 3 โ 4 โ3 โ2 โ1โ1 โ2 FIGURE 1.88 3 โ1 โ2 โ 4 f(x) = 2 โ x 3 FIGURE 1.89 h(x) = 2 โ x 31 8 1 2 3 4 x Compare the graph of each function with the graph of 2x hx f 1 gx f 2x b. a. f x 2 x3. Solution a. Relative to the graph of f x 2 x3, the graph of gx f 2x 2 2x3 2 8x3 is a horizontal shrink c > 1 of the graph of f. (See Figure 1.88.) b. Similarly, the graph of hx f 1 2x 2 1 2x3 2 1 8x3 is a horizontal stretch 0 < c < 1 of the graph of f. (See Figure 1.89.) Now try Exercise 27. 333202_0107.qxd 12/7/05 8:41 AM Page 79 Section 1.7 Transformations of Functions 79 1.7 Exercises VOCABULARY CHECK: In Exercises 1โ5, fill in the blanks. 1. Horizontal shifts, vertical shifts, |
and reflections are called ________ transformations. 2. A reflection in the -axis of x is represented by y f x hx 3. Transformations that cause a distortion in the shape of the graph of is represented by ________. y f x hx of y f x are ________, while a reflection in the -axis y called ________ transformations. 4. A nonrigid transformation of a ________ ________ if 5. A nonrigid transformation of a ________ ________ if y f x 0 < c < 1. y f x 0 < c < 1. represented by hx f cx is a ________ ________ if c > 1 and represented by gx cf x is a ________ ________ if c > 1 and 6. Match the rigid transformation of y f x with the correct representation of the graph of where h, c > 0. (a) (b) (c) (d) hx f x c hx f x c hx f x c hx f x c (i) A horizontal shift of cf, units to the right (ii) A vertical shift of cf, units downward (iii) A horizontal shift of cf, units to the left (iv) A vertical shift of cf, units upward PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1. For each function, sketch (on the same set of coordinate axes) a graph of each function for c 1, 1, and 3. (a) (b) (c. For each function, sketch (on the same set of coordinate axes) a graph of each function for c 3, 1, 1, and 3. (a) (b) (c. For each function, sketch (on the same set of coordinate axes) a graph of each function for c 2, 0, and 2. (a) (b) (c. (a) (b) (c) (d) (e) (f) (g 2x y 6 4 2 (1, 0) โ4 โ2 โ4 (3, 1) f (4, 2) 6 2 4 (0, 1)โ x 6. (a) (b) (c) (d) (e) (f) (g 2x y 8 โ ( 4, 2) โ4 โ โ ( 2, 2) โ6 (6, 2) f x 4 |
(0, 2)โ 8 4. For each function, sketch (on the same set of coordinate axes) a graph of each function for c 3, 1, 1, and 3. FIGURE FOR 5 FIGURE FOR 6 (a) (b) f x x2 c, x2 c, f x x c2, x c2 In Exercises 5โ8, use the graph of to sketch each graph.To print an enlarged copy of the graph go to the website www.mathgraphs.com. f 7. (a) (b) (c) (d) (e) (f) (g 2x 8. (a) (b) (c 10 (f) 3x (g) y f 1 (d) (e) 333202_0107.qxd 12/7/05 8:41 AM Page 80 80 Chapter 1 Functions and Their Graphs y (3, 0) x 11. Use the graph of f x x to write an equation for each function whose graph is shown. (a) y (b) y y 6 2 โ ( 2, 4) f โ4 โ2 โ2 โ4 (0, 3) (1, 0) 6 4 (3, 1)โ x (0, 5) โ3, 0) ( 2 โ10 โ6 โ โ ( 6, 4) โ2 โ6 โ10 โ14 2 6 f (6, 4)โ 4 2 FIGURE FOR 7 FIGURE FOR 8 9. Use the graph of f x x2 to write an equation for each function whose graph is shown. โ2 y (c) 2 4 โ6 x x (d) y โ4 โ6 x x (a) y 2 1 โ2 โ1 1 2 โ2 (c) y 6 4 2 โ2 2 4 6 x x (b) y x 1 โ3 โ1 โ1 โ2 โ3 (d) y 4 2 2 4 6 8 x 10. Use the graph of f x x3 to write an equation for each function whose graph is shown. (a) y 3 2 (b) y 3 2 1 โ2 โ1 โ1 x 2 โ1 1 2 3 (c) y (d) 4 2 โ2 x 2 โ6 โ4 y 4 โ4 โ4 โ8 โ12 4 8 x x 2 4 6 4 8 12 โ4 โ2 โ4 โ6 12. Use the graph of |
f x x to write an equation for each function whose graph is shown. (a) y 4 2 โ2 โ4 โ6 โ8 x 6 8 10 (b) y 2 โ2 โ4 โ8 โ10 2 4 6 8 10 (c) y (d) 8 6 4 2 โ2 โ4 2 4 6 8 10 x y 2 โ4 โ2 2 4 6 โ4 โ8 โ10 x x In Exercises 13โ18, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. 13. y 14. 2 โ2 x 2 4 y 2 โ2 x 2 333202_0107.qxd 12/7/05 8:41 AM Page 81 15. y 16. โ2 x 2 y 6 4 โ2 โ4 17. y 18. 2 โ2 x 4 x 2 4 โ2 โ2 y 4 โ4 โ2 x In Exercises 19โ42, g is related to one of the parent functions described in this chapter. (a) Identify the parent function f. (b) Describe the sequence of tranformations from f to g. (c) Sketch the graph of g. (d) Use function notation to write g in terms of f. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 39. 41. g x 12 x2 g x x3 7 gx 2 3x2 4 g x 2 x 52 gx 3x g x x 13 2x 4 20. 22. 24. 26. 28. 30. 32. 34. 36. 38. 40. 42. g x x 82 g x x 3 1 gx 2x 72 gx x 102 5 gx 1 4 x g x x 33 10 2x 3x 1 In Exercises 43โ50, write an equation for the function that is described by the given characteristics. 43. The shape of f x x2, but moved two units to the right and eight units downward 44. The shape of f x x2 seven units upward, and reflected in the -axis, but moved three units to the left, x 45. The shape of 46. The shape of f x x3 f x x3, but moved 13 units to the right, but moved six units to the left, six 47. The shape of units downward, and reflected in the -axis f x x, x reflected |
in the -axis y but moved 10 units upward and 48. The shape of f x x, seven units downward but moved one unit to the left and Section 1.7 Transformations of Functions 81 49. The shape of f x x, but moved six units to the left and 50. The shape of x reflected in both the -axis and the -axis f x x, but moved nine units downward x and reflected in both the -axis and the -axis y y 51. Use the graph of f x x2 to write an equation for each function whose graph is shown. (a) y 1 โ3 โ2 โ1 1 2 3 x (b) y (1, 7) (1, 3)โ 2 โ5 โ2 2 4 x 52. Use the graph of f x x3 to write an equation for each function whose graph is shown. (a) (b) y 6 4 2 (2, 2) y 3 2 โ6 โ4 2 4 6 x โ3 โ2 โ1 โ4 โ6 โ2 โ3 x 1 3 2 (1, 2)โ 53. Use the graph of f x x to write an equation for each function whose graph is shown. (a) โ4 y 4 2 โ4 โ6 โ8 (4, 2)โ (b) x 6 โ ( 2, 3) y 8 6 4 โ4 โ2 2 4 6 x โ4 54. Use the graph of f x x to write an equation for each function whose graph is shown. (a) y (b) y 20 16 12 8 4 โ4 (4, 16) 4 8 12 16 20 x 1 โ1 โ2 โ3 1 x )โ ( 4, 1 2 333202_0107.qxd 12/7/05 8:41 AM Page 82 82 Chapter 1 Functions and Their Graphs In Exercises 55โ60, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer. 56. 58. 55. 57. y 2 1 โ2 โ1 1 2 x โ2 y 4 2 โ4 โ6 โ8 โ4 x 4 6 59. y 60. y 5 4 โ3 โ2 โ1 1 2 3 y 3 2 1 โ3 โ1 1 2 3 โ2 โ3 y 4 2 x x x x โ6 โ4 โ2 42 6 โ4 โ3 |
โ2 โ1 2 1 โ1 โ2 Graphical Analysis In Exercises 61โ64, use the viewing window shown to write a possible equation for the transformation of the parent function. 61. โ4 63. โ4 6 โ2 1 โ7 62. โ10 64. โ4 8 8 7 โ1 5 โ3 2 8 Graphical Reasoning In Exercises 65 and 66, use the graph of f to sketch the graph of g. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 65. 66. y 4 3 2 f 1 432 5 x โ4 โ3 โ2 โ1 โ2 โ3 (a) (c) (e) gx f x 2 gx f x gx f 4x (b) (d) (f) gx f x 1 gx 2f x gx f 1 2x y 6 4 f โ4 โ2 โ4 โ6 2 4 86 10 12 x (a) (c) (e) gx f x 5 gx f x gx f 2x 1 (b) gx f x 1 2 gx 4 f x 4x 2 (d) (f) gx f 1 Model It 67. Fuel Use The amounts of fuel (in billions of gallons) used by trucks from 1980 through 2002 can be approximated by the function F ft 20.6 0.035t2, 0 โค t โค 22 F t where represents the year, with 1980. corresponding to (Source: U.S. Federal Highway Administration) t 0 (a) Describe the transformation of the parent function Then sketch the graph over the specified f x x2. domain. (b) Find the average rate of change of the function from 1980 to 2002. Interpret your answer in the context of the problem. (c) Rewrite the function so that t 0 Explain how you got your answer. represents 1990. (d) Use the model from part (c) to predict the amount of fuel used by trucks in 2010. Does your answer seem reasonable? Explain. 333202_0107.qxd 12/7/05 2:48 PM Page 83 M 68. Finance The amounts (in trillions of dollars) of mortgage debt outstanding in the United States from 1990 through 2002 can be approximated by the function M f t 0.0054t 20.3962, 0 โค t โค 12 represents the year, with corresponding to (Source: |
Board of Governors of the Federal t 0 t where 1990. Reserve System) (a) Describe the transformation of the parent function Then sketch the graph over the specified f x x2. domain. (b) Rewrite the function so that t 0 represents 2000. Explain how you got your answer. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 69 and 70, determine whether 69. The graphs of f x x 6 are identical. and f x x 6 70. If the graph of the parent function is moved six units to the right, three units upward, and reflected in the x -axis, then the point will lie on the graph of the transformation. 2, 19 f x x2 71. Describing Profits Management originally predicted that the profits from the sales of a new product would be f approximated by the graph of the function shown. The g along with a actual profits are shown by the function verbal description. Use the concepts of transformations of f. graphs to write in terms of g y 40,000 20,000 f (a) The profits were only three-fourths as large as expected. (b) The profits were consistently $10,000 greater than predicted. t 2 4 40,000 20,000 y y g 2 4 60,000 30,000 g 2 4 t t Section 1.7 Transformations of Functions 83 (c) There was a two-year y delay in the introduction of the product. After sales began, profits grew as expected. 40,000 20,000 g t 2 4 6 72. Explain why the graph of y f x is a reflection of the graph of y f x about the -axis. x 73. The graph of y f x 0, 1, passes through the points Find the corresponding points on the 2, 3. 1, 2, graph of and y f x 2 1. 74. Think About It You can use either of two methods to graph a function: plotting points or translating a parent function as shown in this section. Which method of graphing do you prefer to use for each function? Explain. (a) f x 3x2 4x 1 (b) f x 2x 12 6 Skills Review In Exercises 75โ82, perform the operation and simplify. 75. 77. 78. 79. 80. 81. 82. 4 x 4 1 x 76 xx x2 4 x2 x 2 x x2 4 x2 x2 9 |
x 3 5 x x2 3x 28 x2 3x x2 5x 4 In Exercises 83 and 84, evaluate the function at the specified values of the independent variable and simplify. 83. 84. f 3 f x x2 6x 11 (a) f x x 10 3 (a) f 10 (b) (b) f 1 2 f 26 (c) f x 3 (c) f x 10 In Exercises 85โ88, find the domain of the function. 85. f x 2 11 x 87. f x 81 x2 86. f x x 3 x 8 88. f x 34 x2 333202_0108.qxd 12/7/05 8:43 AM Page 84 84 Chapter 1 Functions and Their Graphs 1.8 Combinations of Functions: Composite Functions What you should learn โข Add, subtract, multiply, and divide functions. โข Find the composition of one function with another function. โข Use combinations and compositions of functions to model and solve real-life problems. Why you should learn it Compositions of functions can be used to model and solve real-life problems. For instance, in Exercise 68 on page 92, compositions of functions are used to determine the price of a new hybrid car. ยฉ Jim West/The Image Works Arithmetic Combinations of Functions Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be the functions given by combined to create new functions. For example, f x 2x 3 gx x 2 1 can be combined to form the sum, difference, g. product, and quotient of and and f f x gx 2x 3 x 2 1 x 2 2x 4 f x gx 2x 3 x 2 1 Sum x 2 2x 2 Difference f xgx 2x 3x 2 1 2x 3 3x 2 2x 3 2x 3 x2 1, x ยฑ1 f x gx Product Quotient The domain of an arithmetic combination of functions and real numbers that are common to the domains of and tient there is the further restriction that f g. gx 0. fxgx, g f consists of all In the case of the quo- f Sum, Difference, Product, and Quotient of Functions x Let and be two functions with overlapping domains. Then, for all common to both domains, the sum, difference, product, and quotient of and are |
defined as follows. g g f 1. Sum: 2. Difference: 3. Product: 4. Quotient: f gx f x gx f gx f x gx fgx f x gx x f x f gx, g gx 0 Example 1 Finding the Sum of Two Functions Given f x 2x 1 and gx x 2 2x 1, find f gx. Solution f gx f x gx 2x 1 x 2 2x 1 x 2 4x Now try Exercise 5(a). 333202_0108.qxd 12/7/05 8:43 AM Page 85 Section 1.8 Combinations of Functions: Composite Functions 85 Example 2 Finding the Difference of Two Functions f x 2x 1 Given the difference when and x 2. gx x 2 2x 1, find f gx. Then evaluate Solution The difference of and f f gx f x gx is g 2x 1 x 2 2x 1 x 2 2. When x 2, the value of this difference is f g2 22 2 2. Now try Exercise 5(b). In Examples 1 and 2, both numbers. So, the domains of numbers. Remember that any restrictions on the domains of considered when forming the sum, difference, product, or quotient of and have domains that consist of all real are also the set of all real and must be f and f g f g g and g f g. f Example 3 Finding the Domains of Quotients of Functions Find x f g and x g f for the functions given by f x x and gx 4 x 2. Then find the domains of fg and gf. Solution The quotient of and f g is and the quotient of and is g f g x f x gx g f x gx. The domain of f these domains is is 0, 2. 0, Domain of f g : 0, 2 So, the domains of g and the domain of f g Domain of g f x 0, includes is and 2, 2. g f : 0, 2 fg Note that the domain of but not yields a zero in the denominator, whereas the domain of but not x 2, gf yields a zero in the denominator. because x 0, x 0 because includes x 2 x 2, The intersection of are as follows. Now try Exercise 5(d). 333202_0108.qxd 12/7/05 8:43 AM Page |
86 86 Chapter 1 Functions and Their Graphs Composition of Functions f ห g x g(x) g f f(g(x)) Domain of g FIGURE 1.90 Domain of f The following tables of values help illustrate the composition f gx given in Example 4. x gx gx f gx x f gx Note that the first two tables can be combined (or โcomposedโ) to produce the values given in the third table. Another way of combining two functions is to form the composition of one with the other. For instance, if the composition of with g gx x 1, f x x2 and is f f gx f x 1 x 12. This composition is denoted as f g and reads as โf composed with g.โ Definition of Composition of Two Functions The composition of the function with the function is g f f gx f gx. The domain of in the domain of f g f. is the set of all (See Figure 1.90.) x in the domain of g such that gx is Example 4 Composition of Functions Given f x x 2 a. f gx b. and g f x c. g f 2 gx 4 x2, find the following. Solution a. The composition of with f g is as follows. f gx f gx. The composition of with is as follows. g f g f x g f x gx 2 4 x 22 4 x2 4x 4 x2 4x Definition of f g Definition of Definition of gx f x Simplify. Definition of g f Definition of Definition of f x gx Expand. Simplify. Note that, in this case, f gx g f x. c. Using the result of part (b), you can write the following. g f 2 22 42 4 8 4 Now try Exercise 31. Substitute. Simplify. Simplify. 333202_0108.qxd 12/7/05 8:43 AM Page 87 Te c h n o l o g y You can use a graphing utility to determine the domain of a composition of functions. For the composition in Example 5, enter the function composition as y 9 x22 9. You should obtain the graph shown below. Use the trace feature to determine that the x-coordinates of points on the graph extend from f gx domain of 3 is to 3. So, the 3 โค x โค 3. 1 โ5 5 |
Section 1.8 Combinations of Functions: Composite Functions 87 Example 5 Finding the Domain of a Composite Function f x x2 9 Given find the domain of and f g. gx 9 x2, find the composition f gx. Then Solution f gx f gx f 9 x2 9 x22 9 9 x2 9 x2 From this, it might appear that the domain of the composition is the set of all real is the set of all real numbers. This, however is not true. Because the domain of numbers and the domain of is 3 โค x โค 3. 3 โค x โค 3, the domain of f g is g f Now try Exercise 35. โ10 In Examples 4 and 5, you formed the composition of two given functions. In calculus, it is also important to be able to identify two functions that make up a given composite function. For instance, the function given by h hx 3x 53 is the composition of with where f hx 3x 53 gx3 fgx. g, f x x3 and gx 3x 5. That is, Basically, to โdecomposeโ a composite function, look for an โinnerโ function and is the inner function an โouterโ function. In the function above, and is the outer function. gx 3x 5 f x x3 h Example 6 Decomposing a Composite Function Write the function given by hx 1 x 22 as a composition of two functions. Solution One way to write as a composition of two functions is to take the inner function to be and the outer function to be gx x 2 h f x 1 x2 x2. Then you can write hx 1 x 22 x 22 f x 2 f gx. Now try Exercise 47. 333202_0108.qxd 12/7/05 8:43 AM Page 88 88 Chapter 1 Functions and Their Graphs Application Example 7 Bacteria Count The number N of bacteria in a refrigerated food is given by NT 20T 2 80T 500, 2 โค T โค 14 T where removed from refrigeration, the temperature of the food is given by is the temperature of the food in degrees Celsius. When the food is Tt 4t 2, 0 โค t โค 3 t is the time in hours. (a) Find the composition where meaning in context. (b) Find the time when the bacterial count reaches 2000. and interpret its NTt Solution a. NTt 204t 22 |
804t 2 500 2016t 2 16t 4 320t 160 500 320t 2 320t 80 320t 160 500 320t 2 420 The composite function as a function of the amount of time the food has been out of refrigeration. represents the number of bacteria in the food NTt b. The bacterial count will reach 2000 when Solve this hours. When you equation to find that the count will reach 2000 when solve this equation, note that the negative value is rejected because it is not in the domain of the composite function. 320t 2 420 2000. t 2.2 Now try Exercise 65. W RITING ABOUT MATHEMATICS Analyzing Arithmetic Combinations of Functions a. Use the graphs of and when x 1, 2, 3, 4, 5 gx of f b. Use the graphs of and when x 1, 2, 3, 4, 5 hx of f f g in Figure 1.91 to make a table showing the values, and 6. Explain your reasoning. f h in Figure 1.91 to make a table showing the values, and 6. Explain your reasoning FIGURE 1.91 333202_0108.qxd 12/7/05 8:43 AM Page 89 Section 1.8 Combinations of Functions: Composite Functions 89 1.8 Exercises VOCABULARY CHECK: Fill in the blanks. 1. Two functions and can be combined by the arithmetic operations of ________, ________, ________, g f and _________ to create new functions. 2. The ________ of the function with f g 3. The domain of is all x f g is f gx fgx. in the domain of such that _______ is in the domain of g f. 4. To decompose a composite function, look for an ________ function and an ________ function. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1โ4, use the graphs of hx f gx. go to the website www.mathgraphs.com. to graph To print an enlarged copy of the graph, and g f 1. y 2 3. 4. y 2 g โ2 2 f โ2 โ2 2 x 6 โ2 2 g โ2 In Exercises 5โ12, (c) find (a) f gx, What is the domain of (b) |
f /gx. f gx, f /g? fgx, and (d) f x x 2, f x 2x 5, f x x 2, f x 2x 5, f x x 2 6, 5. 6. 7. 8. 9. 10. f x x2 4, 11. 12, gx x 2 gx 2 x gx 4x 5 gx 4 gx 1 x gx x2 x2 1 gx 1 x2 gx x3 16. 17. 15. 13. 14. and gx x 4. In Exercises 13 โ24, evaluate the indicated function for fx x 2 1 f g2 f g0 f g3t fg6 5 f g f g f g1 f g1 f gt 2 fg6 0 f g 1 g3 fg 5 f 4 20. 19. 24. 22. 18. 23. 21. In Exercises 25 โ28, graph the functions the same set of coordinate axes. f, g, and f g on 25. 26. 27. 28. f x 1 2 x, f x 1 3 x, f x x 2, f x 4 x 2, gx x 1 gx x 4 gx 2x gx x Graphical Reasoning In Exercises 29 and 30, use a graphf g ing utility to graph in the same viewing winand dow. Which function contributes most to the magnitude of the sum when Which function contributes most to the magnitude of the sum when 0 โค x โค 2? f, g, 29. f x 3x, 30. f x x, 2 x > 6? gx x3 10 gx x In Exercises 31โ34, find (a) f g, (b) g f, and (c) f f. 31. 32. 33. f x x2, f x 3x 5, f x 3x 1, 34. f x x3, gx x 1 gx 5 x gx x3 1 gx 1 x 333202_0108.qxd 12/7/05 8:43 AM Page 90 90 Chapter 1 Functions and Their Graphs In Exercises 35โ42, find (a) domain of each function and each composite function. and (b) g f. f g Find the 35. 36. 37. 38. 39. 40. 41. 42. f x x 4, |
f x 3x 5, f x x2 1, f x x23, f x x, f x x 4, f x 1 x, f x 3, x2 1 gx x 2 gx x3 1 gx x gx x6 gx x 6 gx 3 x gx x 3 gx x 1 In Exercises 43โ46, use the graphs of and to evaluate the functions. g f y y = f(x) 4 3 2 1 y = g(x 43. (a) 44. (a) 45. (a) 46. (a) f g3 f g1 f g2 f g1 (b) (b) (b) (b) fg2 fg4 g f 2 g f 3 In Exercises 47โ54, find two functions and f gx hx. (There are many correct answers.) f g such that 47. 49. 51. hx 2x 12 hx 3x 2 4 hx 1 53. hx x 2 x2 3 4 x2 48. 50. 52. 54. hx 1 x3 hx 9 x hx 4 5x 22 hx 27x 3 6x 10 27x 3 55. Stopping Distance The research and development department of an automobile manufacturer has determined that when a driver is required to stop quickly to avoid an accident, the distance (in feet) the car travels during the driverโs reaction time is given by is the speed of the car in miles per hour. The distance (in feet) 15x 2. traveled while the driver is braking is given by Find the function that represents the total stopping distance T. and on the same set of coordinate axes for Graph the functions 0 โค x โค 60. Bx 1 Rx 3 where B,R, 4x, T x 56. Sales From 2000 to 2005, the sales (in thousands of dollars) for one of two restaurants owned by the same parent company can be modeled by R1 R1 480 8t 0.8t 2, t 0, 1, 2, 3, 4, 5 t 0 where period, the sales restaurant can be modeled by R2 represents 2000. During the same six-year (in thousands of dollars) for the second R2 254 0.78t, t 0, 1, 2, 3, 4, 5. (a) Write a function R3 that represents the total sales of the two restaurants owned by the same |
parent company. (b) Use a graphing utility to graph R1, R2, and R3 in the same viewing window. bt t, 57. Vital Statistics Let United States in year of deaths in the United States in year where corresponds to 2000. be the number of births in the represent the number t 0 and let dt t, (a) If pt is the population of the United States in year t, that represents the percent change ct find the function in the population of the United States. c5. (b) Interpret the value of dt and let t, 58. Pets Let in year States in year where ct be the number of dogs in the United States be the number of cats in the United t 0 corresponds to 2000. pt of dogs and cats in the United States. that represents the total number (a) Find the function t, (b) Interpret the value of p5. (c) Let represent the population of the United States in corresponds to 2000. Find and t 0 nt t, year where interpret ht pt nt. 59. Military Personnel The total numbers of Army personnel from (in thousands) and Navy personnel (in thousands) 1990 to 2002 can be approximated by the models At 3.36t2 59.8t 735 N A and Nt 1.95t2 42.2t 603 t where 1990. t 0 represents the year, with (Source: Department of Defense) corresponding to (a) Find and interpret A Nt. Evaluate this function for t 4, 8, and 12. (b) Find and interpret t 4, 8, and 12. for A Nt. Evaluate this function 333202_0108.qxd 12/7/05 8:43 AM Page 91 Section 1.8 Combinations of Functions: Composite Functions 91 60. Sales The sales of exercise equipment (in millions of dollars) in the United States from 1997 to 2003 can be approximated by the function Et 25.95t2 231.2t 3356 E 62. Graphical Reasoning An electronically controlled thermostat in a home is programmed to lower the temperature automatically during the night. The temperature in the house the time in hours on a 24-hour clock (see figure). (in degrees Fahrenheit) is given in terms of T t, and the U.S. population can be approximated by the function Pt 3.02t 252.0 P (in millions) from 1997 to |
2003 corresponding to represents the year, with (Source: National Sporting Goods Association, t 7 t where 1997. U.S. Census Bureau) (a) Find and interpret ht Et Pt. (b) Evaluate the function in part (a) for t 7, 10, and 12. Model It 61. Health Care Costs The table shows the total amounts (in billions of dollars) spent on health services and supplies in the United States (including Puerto Rico) y2, for the years 1995 through 2001. The variables represent out-of-pocket payments, insurance and premiums, and other types of payments, respectively. (Source: Centers for Medicare and Medicaid Services) y1, y3 Year 1995 1996 1997 1998 1999 2000 2001 y1 146.2 152.0 162.2 175.2 184.4 194.7 205.5 y2 329.1 344.1 359.9 382.0 412.1 449.0 496.1 y3 44.8 48.1 52.1 55.6 57.8 57.4 57.8 (a) Use the regression feature of a graphing utility to y1 and quadratic models for find a linear model for y3. y2 Let y2 (b) Find t 5 y3. (c) Use a graphing utility to graph and y1 represent 1995. What does this sum represent? y2, in the same viewing window. y1, y3, and y1 y2 y3 (d) Use the model from part (b) to estimate the total amounts spent on health services and supplies in the years 2008 and 2010. T 80 70 60 50 ) 15 6 12 Time (in hours) 18 21 t 24 T T4 (a) Explain why is a function of t. (b) Approximate T15. (c) The thermostat is reprogrammed to produce a temperHt How does this Tt 1. for which and H ature change the temperature? (d) The thermostat is reprogrammed to produce a temperHow does this Ht Tt 1. H ature change the temperature? for which (e) Write a piecewise-defined function that represents the graph. 63. Geometry A square concrete foundation is prepared as a base for a cylindrical tank (see figure). r x (a) Write the radius of the tank as a function of the length r x of the sides of the square. ( |
b) Write the area A of the circular base of the tank as a function of the radius (c) Find and interpret A rx. r. 333202_0108.qxd 12/7/05 8:43 AM Page 92 92 Chapter 1 Functions and Their Graphs 64. Physics A pebble is dropped into a calm pond, causing ripples in the form of concentric circles (see figure). The t radius where is the time in seconds after the pebble strikes the water. The Ar r 2. area Find and interpret of the circle is given by the function (in feet) of the outer ripple is rt 0.6t, A rt. A r 65. Bacteria Count The number N of bacteria in a refriger- ated food is given by NT 10T 2 20T 600, 1 โค T โค 20 T where is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by Tt 3t 2, 0 โค t โค 6 where t is the time in hours. (a) Find the composition NTt and interpret its meaning in context. (b) Find the time when the bacterial count reaches 1500. 66. Cost The weekly cost C of producing units in a manu- x facturing process is given by Cx 60x 750. The number of units produced in hours is given by xt 50t. x t (a) Find and interpret C xt. (b) Find the time that must elapse in order for the cost to increase to $15,000. 67. Salary You are a sales representative for a clothing manufacturer. You are paid an annual salary, plus a bonus of 3% of your sales over $500,000. Consider the two functions given by f x x 500,000 g(x) 0.03x. and x is greater than $500,000, which of the following repre- If sents your bonus? Explain your reasoning. (a) f gx (b) g f x 68. Consumer Awareness The suggested retail price of a new hybrid car is dollars. The dealership advertises a factory rebate of $2000 and a 10% discount. p (a) Write a function R in terms of giving the cost of the p hybrid car after receiving the rebate from the factory. (b) Write a function S in terms of giving the cost of the p hybrid car after receiving the dealership discount. (c) |
Form the composite functions R Sp and S Rp and interpret each. R S20,500 (d) Find and S R20,500. Which yields the lower cost for the hybrid car? Explain. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 69 and 70, determine whether 69. If f x x 1 and f g)x g f )x. gx 6x, then 70. If you are given two functions f gx calculate f. of the domain of if and only if the range of f x and gx g, you can is a subset 71. Proof Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function. 72. Conjecture Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis. Skills Review Average Rate of Change difference quotient f x h f x h and simplify your answer. 73. 75. f x 3x 4 f x 4 x In Exercises 73โ76, find the 74. f x 1 x2 76. f x 2x 1 In Exercises 77โ80, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line. 2, 4, 8, 1, 6, 3, 7, 0 77. 80. 79. 78. 333202_0109.qxd 12/7/05 8:45 AM Page 93 1.9 Inverse Functions Section 1.9 Inverse Functions 93 What you should learn โข Find inverse functions informally and verify that two functions are inverse functions of each other. โข Use graphs of functions to determine whether functions have inverse functions. โข Use the Horizontal Line Test to determine if functions are one-to-one. โข Find inverse functions alge- braically. Why you should learn it Inverse functions can be used to model and solve real-life problems. For instance, in Exercise 80 on page 101, an inverse function can be used to determine the year in which there was a given dollar amount of sales of digital cameras in the United States. Inverse Functions Recall from Section 1.4, that a function can be represented by a set of ordered pairs. For instance, the function to the set A 1, 2, 3, 4 from the set B 5, 6, 7, 8 fx x 4: f |
x x 4 can be written as follows. 1, 5, 2, 6, 3, 7, 4, 8 In this case, by interchanging the first and second coordinates of each of these f 1. ordered pairs, you can form the inverse function of which is denoted by It is a function from the set f 1x x 4: to the set 5, 1, 6, 2, 7, 3, 8, 4 and can be written as follows. A, B f, is equal to the range of f Note that the domain of Figure 1.92. Also note that the functions and each other. In other words, when you form the composition of with composition of and vice versa, as shown in have the effect of โundoingโ or the you obtain the identity function. f 1 f 1 f f f 1, f 1 with f, f f 1x 1x 4 x 4 4 x Domain of f Range of f f (x) = x + 4 x Range of f โ1 FIGURE 1.92 โ1(x) = x โ 4 f f(x) Domain of f โ1 ยฉ Tim Boyle/Getty Images Example 1 Finding Inverse Functions Informally Find the inverse function of f 1 f x are equal to the identity function. f(x) 4x. Then verify that both f f 1x and Solution The function multiplies each input by 4. To โundoโ this function, you need to divide each input by 4. So, the inverse function of f x 4x is f f 1x x 4. You can verify that both f f 1x fx 4 4x 4 f f 1x x x Now try Exercise 1. as follows. f 1 f x x and f 1 f x f 14x 4x 4 x 333202_0109.qxd 12/7/05 8:45 AM Page 94 94 Chapter 1 Functions and Their Graphs Exploration Consider the functions given by x 2 f x and f 1x x 2. f f 1x Evaluate f 1 f x for the indicated x. values of What can you conclude about the functions? and 10 0 7 45 x f f1x f1 f x Definition of Inverse Function Let and be two functions such that f g f gx x for every x in the domain of g and g f x x for every x in the domain of f. Under these conditions, the function f. tion The function is denoted by f |
f 1x x g f 1 f 1 f x x. and g is the inverse function of the func- (read โ -inverseโ). So, f f The domain of must be equal to the range of be equal to the domain of f 1. f 1, and the range of must f In this is written, it always refers to the inverse function of the func- to denote the inverse function f 1. 1 Donโt be confused by the use of f 1 text, whenever tion and not to the reciprocal of f If the function f x. f, is the inverse function of the function it must also be true g. that the function is the inverse function of the function For this reason, you can say that the functions and are inverse functions of each other. g g f f Example 2 Verifying Inverse Functions Which of the functions is the inverse function of gx x 2 5 hx 5 x 2 fx 5 x 2? Solution By forming the composition of with f g, you have f gx f x 2 5 5 x 2 5 25 x 12 Substitute x 2 5 for x. 2 x. g Because this composition is not equal to the identity function is not the inverse function of By forming the composition of with you have it follows that x, f h, f. f hx f5. So, it appears that that the composition of with is the inverse function of You can confirm this by showing is also equal to the identity function. f. h h f Now try Exercise 5. 333202_0109.qxd 12/7/05 8:45 AM Page 95 y y = x The Graph of an Inverse Function Section 1.9 Inverse Functions 95 y = f (x) (a, b) y = f โ1(x) (b, a) x x FIGURE 1.93 f โ1( ) = ( + 3โ y 6 (1, 2) (3, 3) (2, 1) (1, 1)โ 6 (0, 3)โ โ ( 1, 1) โ ( 3, 0) โ6 โ โ ( 5, 1) y x= โ โ ( 1, 5) FIGURE 1.94 y (3, 9) f (x) = x2 (2, 4) (4, 2) y = x (9, 3) (1, 1) โ1 (x The graphs of a function and its inverse function |
in the following way. If the point must lie on the graph of a reflection of the graph of are related to each other b, a then the point f 1 is and vice versa. This means that the graph of in the line as shown in Figure 1.93. lies on the graph of f 1, f y x, a, b f, f 1 Example 3 Finding Inverse Functions Graphically x 3 Sketch the graphs of the inverse functions on the same rectangular coordinate system and show that the graphs are reflections of each other in the line f x 2x 3 f 1x 1 2 y x. and f Solution are shown in Figure 1.94. It appears that the graphs are The graphs of and y x. reflections of each other in the line You can further verify this reflective property by testing a few points on each graph. Note in the following list that if the point the point a, b f 1. f 1 f, is on the graph of fx 2x 3 Graph of Graph of is on the graph of x 3 1, 5 0, 3 1, 1 2, 1 3, 3 b, a f 1x 1 2 5, 1 3, 0 1, 1 1, 2 3, 3 Now try Exercise 15. Example 4 Finding Inverse Functions Graphically Sketch the graphs of the inverse functions on the same rectangular coordinate system and show that the graphs are reflections of each other in the line y x. and f x x 2 x โฅ 0 f 1x x f Solution are shown in Figure 1.95. It appears that the graphs are The graphs of and y x. reflections of each other in the line You can further verify this reflective property by testing a few points on each graph. Note in the following list that if the point the point is on the graph of b, a a, b f 1. f 1 f, Graph of x โฅ 0 is on the graph of fx x 2, 0, 0 1, 1 2, 4 3, 9 Graph of f 1x x 0, 0 1, 1 4, 2 9, 3 (0, 0) 3 4 5 6 7 8 9 x Try showing that f f 1x x and f 1 f x x. FIGURE 1.95 Now try Exercise 17. 333202_0109.qxd 12/7/05 8:46 AM Page 96 96 Chapter 1 Functions and Their Graphs One-to-One Functions The reflective property of the graphs of inverse functions gives you a nice geometric test for |
determining whether a function has an inverse function. This test is called the Horizontal Line Test for inverse functions. Horizontal Line Test for Inverse Functions A function has an inverse function if and only if no horizontal line f intersects the graph of at more than one point. f If no horizontal line intersects the graph of at more than one point, then no y x -value is matched with more than one -value. This is the essential characteristic of what are called one-to-one functions. f One-to-One Functions A function sponds to exactly one value of the independent variable. A function has an is one-to-one. inverse function if and only if is one-to-one if each value of the dependent variable corre- f f f f x x2. f x x2. Consider the function given by The table on the left is a table of values for The table of values on the right is made up by interchanging the columns of the first table. The table on the right does not represent a funcand tion because the input y 2. x 4 y 2 is not one-to-one and does not have an inverse function. is matched with two different outputs: f x x2 So x2 Example 5 Applying the Horizontal Line Test a. The graph of the function given by f x x3 1 Because no horizontal line intersects the graph of you can conclude that function. is shown in Figure 1.96. f at more than one point, is a one-to-one function and does have an inverse f b. The graph of the function given by is shown in Figure 1.97. Because it is possible to find a horizontal line that intersects the graph of at is not a one-to-one function and more than one point, you can conclude that does not have an inverse function. f f fx x 2 1 Now try Exercise 29. y 3 1 โ2 โ3 y 3 2 โ2 โ3 โ3 โ2 โ1 FIGURE 1.96 โ3 โ2 FIGURE 1.97 x 2 3 f (xx) = x 2 โ 1 333202_0109.qxd 12/7/05 8:46 AM Page 97 Section 1.9 Inverse Functions 97 Finding Inverse Functions Algebraically Note what happens when you try to find the inverse function of a function that is not one-to-one. f x x2 1 y x2 1 x y2 1 x 1 |
y2 Original function Replace f(x) by y. Interchange x and y. Isolate y-term. For simple functions (such as the one in Example 1), you can find inverse functions by inspection. For more complicated functions, however, it is best to use the following guidelines. The key step in these guidelines is Step 3โinterchanging the roles of and This step corresponds to the fact that inverse functions have ordered pairs with the coordinates reversed. y. x Finding an Inverse Function 1. Use the Horizontal Line Test to decide whether has an inverse function. f x 2. In the equation for replace f x, by y. f 3. Interchange the roles of and x y, and solve for y. y ยฑ x 1 Solve for y. 4. Replace by y f 1x in the new equation. You obtain two -values for each x. y f x( ) = 5 โ 3x 2 x 4 6 y 6 4 โ2 โ 4 โ6 โ6 โ4 โ2 FIGURE 1.98 Exploration Restrict the domain of f x x2 1 Use to a graphing utility to graph the function. Does the restricted function have an inverse function? Explain. x โฅ 0. f 5. Verify that and f the domain of f 1, domain of are inverse functions of each other by showing that is equal to the f 1 is equal to the range of and f f 1x x and f 1 f x x. the range of f 1, f Example 6 Finding an Inverse Function Algebraically Find the inverse function of f x 5 3x. 2 is a line, as shown in Figure 1.98. This graph passes the Horizontal f is one-to-one and has an inverse function. Solution f The graph of Line Test. So, you know that f x 5 3x 2 y 5 3x 2 x 5 3y 2 2x 5 3y 3y 5 2x y 5 2x 3 f 1x 5 2x 3 Write original function. Replace f x by y. Interchange and x y. Multiply each side by 2. Isolate the -term. y Solve for y. Replace by y f 1x. f 1 Note that both and real numbers. Check that f have domains and ranges that consist of the entire set of f f 1x x f 1 f x x. and Now try Exercise 55. 333202_0109.qxd 12/7/05 8:46 AM Page |
98 98 Chapter 1 Functions and Their Graphs Example 7 Finding an Inverse Function Find the inverse function of f x 3x 13 โ2 1 2 3 x โ1 โ2 โ3 FIGURE 1.99 Solution The graph of Horizontal Line Test, you know that f is a curve, as shown in Figure 1.99. Because this graph passes the is one-to-one and has an inverse function. f f x 3x 1 y 3x 1 x 3y 1 x3 y 1 x3 1 y x 3 1 f 1x f 1 f Write original function. Replace f x by y. Interchange and x y. Cube each side. Solve for y. Replace by y f 1x. and Both numbers. You can verify this result numerically as shown in the tables below. have domains and ranges that consist of the entire set of real x 28 9 2 1 0 7 26 f x 3 2 1 0 1 2 3 Now try Exercise 61 1x 28 9 2 1 0 7 26 W RITING ABOUT MATHEMATICS The Existence of an Inverse Function Write a short paragraph describing why the following functions do or do not have inverse functions. x f x represent the retail price of an item (in dollars), and represent the sales tax on the item. Assume that a. Let let the sales tax is 6% of the retail price and that the sales tax is rounded to the nearest cent. Does this function have an inverse function? (Hint: Can you undo this function? For instance, if you know that the sales tax is $0.12, can you determine exactly what the retail price is?) x f x represent the temperature in degrees Celsius, and represent the temperature in degrees Fahrenheit. b. Let let Does this function have an inverse function? (Hint: The formula for converting from degrees Celsius to degrees ) Fahrenheit is F 9 5 C 32. 333202_0109.qxd 12/7/05 8:46 AM Page 99 1.9 Exercises Section 1.9 Inverse Functions 99 g is the ________ function of f. VOCABULARY CHECK: Fill in the blanks. 1. If the composite functions and 2. The domain of f fgx x is the ________ of f 1, g fx x and the ________ of then the function f 1 is the range of f. 3. The graphs of and f f 1 are reflections of each other in the line ________ |
. 4. A function f is ________ if each value of the dependent variable corresponds to exactly one value of the independent variable. 5. A graphical test for the existence of an inverse function of f is called the _______ Line Test. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1โ 8, find the inverse function of Verify that f 1f x x. f f 1x x and f informally. 11. y 12. 1. 3. 5. 7. f x 6x f x x 9 f x 3x 1 f x 3x 2. 4. 6. 8 x5 In Exercises 9โ12, match the graph of the function with the graph of its inverse function. [The graphs of the inverse functions are labeled (a), (b), (c), and (d).] (a) y (b) y 4 3 2 1 (c) 91 1 2 3 โ2 y 4 3 2 1 โ2 โd) y 3 2 1 โ2 โ3 1 2 3 โ3 โ2 103 โ3 f In Exercises 13โ24, show that and are inverse functions (a) algebraically and (b) graphically. g 13. f x 2x, 14. f x x 5, 15. f x 7x 1, 16. f x 3 4x, 17. 18. 19. 20. 21. 22. 23. 24., f x x3, f x 1 x3, f x 9 x 2, x โฅ 0, gx x 2 gx x 5 gx x 1 7 gx 3 x 4 gx 38x gx 1 x gx x2 4, gx 31 x gx 9 x, gx 1 x x gx 5x 1 x 1 gx 2x 3 x 1, 333202_0109.qxd 12/7/05 8:46 AM Page 100 100 Chapter 1 Functions and Their Graphs In Exercises 25 and 26, does the function have an inverse function? 25. 26 10 10 In Exercises 27 and 28, use the table of values for to complete a table for y f 1x. y f x 27. x f x 28. x 2 2 3 f x 10 In Exercises 29โ32, does the function have an inverse function? 29. y 30. y 6 4 |
2 โ2 2 4 6 31. y 2 โ2 2 โ2 6 2 x x โ4 โ2 โ2 2 4 32. y 4 2 โ2 2 4 6 x x In Exercises 33โ38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. 34. f x 10 gx 4 x 6 hx x 4 x 4 gx x 53 f x 2x16 x2 f x 1 x 22 1 8 33. 35. 36. 37. 38. f, In Exercises 39โ54, (a) find the inverse function of on the same set of coordinate axes, (b) graph both and (c) describe the relationship between the graphs of and 42. 40., and (d) state the domain and range of and f f x 2x 3 f x 3x 1 f x x5 2 f x x3 1 f x x f x x 2, f x 4 x2, f x x2 2 44 50. 48. f x 3x 1 f x 6x 4 4x 5 52. 54. f x x35 f x 8x 4 2x 6 39. 41. 43. 45. 46. 47. 49. 51. 53. In Exercises 55โ68, determine whether the function has an inverse function. If it does, find the inverse function. 55. f x x4 57. gx x 8 59. px 4 56. f x 1 x 2 58. f x 3x 5 60. f x 3x 4 5 x โฅ 3 62. x < 0 x โฅ 0 64. qx x 52 f x x, x2 3x, x โค 0 x > 0 61. 63. 65. f x x 32, f x x 3, 6 x, hx 4 x2 y 1 1 โ1 โ2 67. f x 2x 3 y 4 3 2 1 โ4 โ3 โ2 โ1 1 2 66. f x x 2, x โค 2 y 4 2 1 โ1 โ2 โ3 โ4 1 2 3 4 5 6 682 โ2 โ2 333202_0109.qxd 12/7/05 8:46 AM Page 101 Section 1.9 Inverse Functions 101 In Exercises 69โ74, use the f x 1 and function. gx x 3 8x 3 |
functions given by to find the indicated value or 69. 71. 73. f 1 g11 f 1 f 16 ( f g)1 70. 72. 74. g1 f 13 g1 g14 g1 f 1 80. Digital Camera Sales The factory sales (in millions of dollars) of digital cameras in the United States from 1998 through 2003 are shown in the table. The time (in years) is given by with (Source: Consumer Electronincs Association) corresponding to 1998. t 8 t, f f x x 4 Year, t Sales, f t In Exercises 75โ78, use the functions given by and to find the specified function. gx 2x 5 75. 77. f 1 g1 f g1 76. 78. f 1 g1 g f 1 Model It 79. U.S. Households The numbers of households (in thousands) in the United States from 1995 to 2003 are t, shown in the table. The time (in years) is given by t 5 (Source: U.S. with corresponding to 1995. Census Bureau) f 8 9 10 11 12 13 519 1209 1825 1972 2794 3421 exist? f 1 exists, what does it represent in the context of the (b) If (a) Does f 1 problem? f 1 (c) If exists, find f 11825. Year, t Households, f t (d) If the table was extended to 2004 and if the factory sales of digital cameras for that year was $2794 exist? Explain. million, would f1 5 6 7 8 9 10 11 12 13 98,990 99,627 101,018 102,528 103,874 104,705 108,209 109,297 111,278 (a) Find f 1108,209. f 1 (b) What does mean in the context of the problem? (c) Use the regression feature of a graphing utility to y mx b. find a linear model for the data, (Round to two decimal places.) and m b (d) Algebraically find the inverse function of the linear model in part (c). (e) Use the inverse function of the linear model you found in part (d) to approximate f 1117, 022. (f) Use the inverse function of the linear model you found in part (d) to approximate How does this value compare with the original data shown in the table? f1108,209. 81. Miles |
Traveled The total numbers (in billions) of miles traveled by motor vehicles in the United States from 1995 through 2002 are shown in the table. The time t, (in years) is given by with corresponding to 1995. (Source: U.S. Federal Highway Administration) t 5 f Year, t Miles traveled, f t 5 6 7 8 9 10 11 12 2423 2486 2562 2632 2691 2747 2797 2856 exist? f 1 exists, what does it mean in the context of the (b) If (a) Does f 1 problem? f 1 (c) If exists, find f 12632. (d) If the table was extended to 2003 and if the total number of miles traveled by motor vehicles for that year was 2747 billion, would exist? Explain. f 1 333202_0109.qxd 12/7/05 8:46 AM Page 102 102 Chapter 1 Functions and Their Graphs 82. Hourly Wage Your wage is $8.00 per hour plus $0.75 for each unit produced per hour. So, your hourly wage in terms of the number of units produced is y y 8 0.75x. (a) Find the inverse function. (b) What does each variable represent in the inverse function? (c) Determine the number of units produced when your hourly wage is $22.25. 83. Diesel Mechanics The function given by y 0.03x 2 245.50, 0 < x < 100 approximates the exhaust temperature Fahrenheit, where in degrees is the percent load for a diesel engine. x y (a) Find the inverse function. What does each variable represent in the inverse function? (b) Use a graphing utility to graph the inverse function. (c) The exhaust temperature of the engine must not exceed 500 degrees Fahrenheit. What is the percent load interval? 84. Cost You need a total of 50 pounds of two types of ground beef costing $1.25 and $1.60 per pound, respectively. A model for the total cost of the two types of beef is y 1.25x 1.6050 x y x where ground beef. is the number of pounds of the less expensive (a) Find the inverse function of the cost function. What does each variable represent in the inverse function? (b) Use the context of the problem to determine the domain of the inverse function. (c) Determine the number of pounds of the less expensive ground beef purchased when the total cost |
is $73. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 85 and 86, determine whether 85. If f is an even function, exists. 86. If the inverse function of exists and the graph of f f 1. is an -intercept of y y -intercept, the -intercept of x f has a f 1 f In Exercises 89โ 92, use the graph of the function to create a table of values for the given points. Then create a second table that can be used to find, and sketch the graph of f 1 if possible. f 1 f 89. y 90 91. 4 f โ2 โ4 โ2 2 4 x 92. y f x 4 โ4 โ2 โ2 โ4 93. Think About It The function given by f x k2 x x 3 has an inverse function, and f 1(3) 2. Find k. 94. Think About It The function given by f x kx3 3x 4 has an inverse function, and f 1(5) 2. Find k. Skills Review In Exercises 95โ102, solve the equation using any convenient method. 95. 96. 97. 98. 99. 100. 101. 102. x 2 64 x 52 8 4x 2 12x 9 0 9x 2 12x 3 0 x 2 6x 4 0 2x 2 4x 6 0 50 5x 3x 2 2x 2 4x 9 2x 12 87. Proof Prove that if and are one-to-one functions, then g f g1x g1 f 1x. f 88. Proof Prove that if f is an odd function. f 1 is a one-to-one odd function, then 103. Find two consecutive positive even integers whose product is 288. 104. Geometry A triangular sign has a height that is twice its base. The area of the sign is 10 square feet. Find the base and height of the sign. 333202_0110.qxd_pg 103 1/9/06 8:52 AM Page 103 Section 1.10 Mathematical Modeling and Variation 103 1.10 Mathematical Modeling and Variation What you should learn โข Use mathematical models to approximate sets of data points. โข Use the regression feature of a graphing utility to find the equation of a least squares regression line. โข Write mathematical models for direct variation. โข Write mathematical models for direct variation as an |
nth power. โข Write mathematical models for inverse variation. โข Write mathematical models for joint variation. Why you should learn it You can use functions as models to represent a wide variety of real-life data sets. For instance, in Exercise 71 on page 113, a variation model can be used to model the water temperature of the ocean at various depths. U.S. Banks y y = โ0.283t + 11.14 11 10 ( Introduction You have already studied some techniques for fitting models to data. For instance, in Section 1.3, you learned how to find the equation of a line that passes through two points. In this section, you will study other techniques for fitting models to data: least squares regression and direct and inverse variation. The resulting models are either polynomial functions or rational functions. (Rational functions will be studied in Chapter 2.) Example 1 A Mathematical Model The numbers of insured commercial banks for the years 1996 to 2001 are shown in the table. Insurance Corporation) y (in thousands) in the United States (Source: Federal Deposit Year 1996 1997 1998 1999 2000 2001 Insured commercial banks, y 9.53 9.14 8.77 8.58 8.32 8.08 t y 0.283t 11.14 where is the year, with for A linear model that approximates the data is t 6 6 โค t โค 11, corresponding to 1996. Plot the actual data and the model on the same graph. How closely does the model represent the data? Solution The actual data are plotted in Figure 1.100, along with the graph of the linear model. From the graph, it appears that the model is a โgood fitโ for the actual data. You can see how well the model fits by comparing the actual values of with y* the values of given by the model. The values given by the model are labeled in the table below. y y t y 6 7 8 9 10 11 9.53 9.14 8.77 8.58 8.32 8.08 y* 9.44 9.16 8.88 8.59 8.31 8.03 6 FIGURE 1.100 7 10 8 Year (6 โ 1996) 9 t 11 Now try Exercise 1. Note in Example 1 that you could have chosen any two points to find a line that fits the data. However, the given linear model was found using the regression feature of a graphing utility and is the line that best fits the data. This concept of |
a โbest-fittingโ line is discussed on the next page. 333202_0110.qxd 12/7/05 2:49 PM Page 104 104 Chapter 1 Functions and Their Graphs Least Squares Regression and Graphing Utilities So far in this text, you have worked with many different types of mathematical models that approximate real-life data. In some instances the model was given (as in Example 1), whereas in other instances you were asked to find the model using simple algebraic techniques or a graphing utility. To find a model that approximates the data most accurately, statisticians use a measure called the sum of square differences, which is the sum of the squares of the differences between actual data values and model values. The โbestfittingโ linear model, called the least squares regression line, is the one with the least sum of square differences. Recall that you can approximate this line visually by plotting the data points and drawing the line that appears to fit bestโor you can enter the data points into a calculator or computer and use the linear regression feature of the calculator or computer. When you use the regression feauture of a graphing calculator or computer program, you will notice that the program may also output an โ -value.โ This -value is the correlation coefficient of the data and gives a measure of how well the model fits the data. The closer the value of is to 1, the better the fit. r r r Example 2 Finding a Least Squares Regression Line p The amounts (in millions of dollars) of total annual prize money awarded at the Indianapolis 500 race from 1995 to 2004 are shown in the table. Construct a scatter plot that represents the data and find the least squares regression line for the data. (Source: indy500.com) Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Prize money, p 8.06 8.11 8.61 8.72 9.05 9.48 9.61 10.03 10.15 10.25 Solution t 5 represent 1995. The scatter plot for the points is shown in Figure 1.101. Let Using the regression feature of a graphing utility, you can determine that the equation of the least squares regression line is p 0.268t 6.66. To check this model, compare the actual -values with the -values given by the model, which are labeled in the table at the left. The correlation coefficient for this model is which implies that the |
model is a good fit. p* r 0.991, p p Now try Exercise 7. Indianapolis 500 p 11 10 FIGURE 1.101 5 6 7 8 9 10 11 12 13 14 Year (5 โ 1995) t t 5 6 7 8 9 10 11 12 13 14 p p* 8.06 8.11 8.61 8.72 9.05 9.48 9.61 10.03 10.15 10.25 8.00 8.27 8.54 8.80 9.07 9.34 9.61 9.88 10.14 10.41 333202_0110.qxd 12/7/05 8:47 AM Page 105 Section 1.10 Mathematical Modeling and Variation 105 Direct Variation There are two basic types of linear models. The more general model has a y -intercept that is nonzero. b 0 y mx b, The simpler model y kx y has a -intercept that is zero. In the simpler model, x, or to be directly proportional to x. y is said to vary directly as Direct Variation The following statements are equivalent. 1. y varies directly as x. 2. 3. y y kx is directly proportional to x. for some nonzero constant k. k is the constant of variation or the constant of proportionality. Example 3 Direct Variation In Pennsylvania, the state income tax is directly proportional to gross income. You are working in Pennsylvania and your state income tax deduction is $46.05 for a gross monthly income of $1500. Find a mathematical model that gives the Pennsylvania state income tax in terms of gross income. Solution Verbal Model: Labels: Equation: State income tax k Gross income State income tax y Gross income x Income tax rate k y kx (dollars) (dollars) (percent in decimal form) substitute the given information into the equation y kx, and then k, To solve for k. solve for y kx 46.05 k1500 0.0307 k Write direct variation model. Substitute y 46.05 and x 1500. Simplify. So, the equation (or model) for state income tax in Pennsylvania is x y 0.0307x. In other words, Pennsylvania has a state income tax rate of 3.07% of gross income. The graph of this equation is shown in Figure 1.102. Now try Exercise 33. Pennsylvania Taxes y y = 0.0307x (1500 |
, 46.05) 100 80 60 40 20 ) 1000 2000 3000 4000 Gross income (in dollars) FIGURE 1.102 333202_0110.qxd 12/7/05 8:47 AM Page 106 106 Chapter 1 Functions and Their Graphs Direct Variation as an nth Power Another type of direct variation relates one variable to a power of another variable. For example, in the formula for the area of a circle A r2 the area formula, A is directly proportional to the square of the radius Note that for this is the constant of proportionality. r. Note that the direct variation model of with n 1. is a special case y kx n y kx Direct Variation as an nth Power The following statements are equivalent. 1. y varies directly as the nth power of x. is directly proportional to the nth power of x. 2. 3. y y kx n for some constant k. t = 0 sec t = 1 sec 10 20 30 FIGURE 1.103 Example 4 Direct Variation as nth Power The distance a ball rolls down an inclined plane is directly proportional to the square of the time it rolls. During the first second, the ball rolls 8 feet. (See Figure 1.103.) 40 50 60 t = 3 sec 70 a. Write an equation relating the distance traveled to the time. b. How far will the ball roll during the first 3 seconds? Solution a. Letting be the distance (in feet) the ball rolls and letting be the time (in t d seconds), you have d kt 2. when t 1, you can see that k 8, as follows. d 8 Now, because d kt 2 8 k12 8 k So, the equation relating distance to time is d 8t 2. t 3, b. When the distance traveled is d 832 89 72 feet. Now try Exercise 63. d 1 In Examples 3 and 4, the direct variations are such that an increase in one variable corresponds to an increase in the other variable. This is also true in the model results in an increase in You should not, however, assume that this always occurs with direct variation. For example, in the model and an increase in yet y 3x, is said to vary directly as x. results in a decrease in where an increase in 5F, F > 0, d. y, F x y 333202_0110.qxd 12/7/05 8:47 AM Page 107 Section 1. |
10 Mathematical Modeling and Variation 107 Inverse Variation Inverse Variation The following statements are equivalent. 1. 3. x. 2. y is inversely proportional to x. y varies inversely as y k x for some constant k. x y If and are related by an equation of the form as the th power of (or n y x is inversely proportional to the th power of ). y then varies inversely n x y kx n, Some applications of variation involve problems with both direct and inverse variation in the same model. These types of models are said to have combined variation. Example 5 Direct and Inverse Variation P1 V1 A gas law states that the volume of an enclosed gas varies directly as the temperature and inversely as the pressure, as shown in Figure 1.104. The pres294 sure of a gas is 0.75 kilogram per square centimeter when the temperature is K and the volume is 8000 cubic centimeters. (a) Write an equation relating pressure, temperature, and volume. (b) Find the pressure when the temperature is 300 K and the volume is 7000 cubic centimeters. P2 V2 P2 > P1 then < V2 V1 If the temperature is held FIGURE 1.104 constant and pressure increases, volume decreases. Solution V a. Let be volume (in cubic centimeters), let P be pressure (in kilograms per varies V be temperature (in Kelvin). Because P, you have T and inversely as square centimeter), and let T directly as V kT P. Now, because when T 294 and V 8000, you have P 0.75 8000 k294 0.75 k 6000 294 1000 49. So, the equation relating pressure, temperature, and volume is b. When the pressure is. T P V 1000 49 T 300 P 1000 49 and 300 7000 V 7000, 300 343 Now try Exercise 65. 0.87 kilogram per square centimeter. 333202_0110.qxd 12/7/05 8:47 AM Page 108 108 Chapter 1 Functions and Their Graphs Joint Variation In Example 5, note that when a direct variation and an inverse variation occur in the same statement, they are coupled with the word โand.โ To describe two different direct variations in the same statement, the word jointly is used. Joint Variation The following statements are equivalent. 1. z varies jointly as and x y. is jointly proportional to and x y. for some constant k. z z kxy |
2. 3. If x, and are related by an equation of the form z y, z kx ny m then varies jointly as the th power of and the n x z m th power of y. Example 6 Joint Variation The simple interest for a certain savings account is jointly proportional to the time and the principal. After one quarter (3 months), the interest on a principal of $5000 is $43.75. a. Write an equation relating the interest, principal, and time. b. Find the interest after three quarters. Solution a. Let I interest (in dollars), P is jointly proportional to principal (in dollars), and t, and you have P t time (in I years). Because I kPt. I 43.75, P 5000, 43.75 k50001 4 For and t 1 4, you have which implies that interest, principal, and time is k 443.755000 0.035. So, the equation relating I 0.035Pt which is the familiar equation for simple interest where the constant of proportionality, 0.035, represents an annual interest rate of 3.5%. b. When P $5000 and I 0.03550003 4 t 3 4, the interest is $131.25. Now try Exercise 67. 333202_0110.qxd 12/7/05 8:47 AM Page 109 Section 1.10 Mathematical Modeling and Variation 109 1.10 Exercises VOCABULARY CHECK: Fill in the blanks. 1. Two techniques for fitting models to data are called direct ________ and least squares ________. 2. Statisticians use a measure called ________ of________ ________ to find a model that approximates a set of data most accurately. 3. An -value of a set of data, also called a ________ ________, gives a measure of how well a model fits a set of data. r 4. Direct variation models can be described as varies directly as x, or y is ________ ________ to x. 5. In direct variation models of the form k is called the ________ of ________. y y kx, 6. The direct variation model y kxn or y is ________ ________ to the th power of y k x 7. The mathematical model is an example of ________ variation. can be described as varies directly as the th power of n x. n y x, 8. Mathematical models that involve both direct |
and inverse variation are said to have ________ variation. 9. The joint variation model z kxy x y. is ________ ________ to and can be described as varies jointly as and x z y, or z PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1998, 137,673 1999, 139,368 2000, 142,583 2001, 143,734 2002, 144,683 1. Employment The total numbers of employees (in thousands) in the United States from 1992 to 2002 are given by the following ordered pairs. 1992, 128,105 1993, 129,200 1994, 131,056 1995, 132,304 1996, 133,943 1997, 136,297 A is linear model y 1767.0t 123,916, represents the number of where employees (in thousands) and represents 1992. Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? (Source: U.S. Bureau of Labor Statistics) that approximates y t 2 the data 2. Sports The winning times (in minutes) in the womenโs 400-meter freestyle swimming event in the Olympics from 1948 to 2004 are given by the following ordered pairs. 1948, 5.30 1952, 5.20 1956, 4.91 1960, 4.84 1964, 4.72 1968, 4.53 1972, 4.32 1976, 4.16 1980, 4.15 1984, 4.12 1988, 4.06 1992, 4.12 1996, 4.12 2000, 4.10 2004, 4.09 y the data A is that approximates linear model y 0.022t 5.03, where represents the winning time t 0 (in minutes) and represents 1950. Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: The World Almanac and Book of Facts) In Exercises 3โ 6, sketch the line that you think best approximates the data in the scatter plot.Then find an equation of the line. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 3. y 4 333202_0110.qxd 12/7/05 8:47 AM Page 110 110 Chapter 1 Functions |
and Their Graphs 7. Sports The lengths (in feet) of the winning menโs discus throws in the Olympics from 1912 to 2004 are listed below. (Source: The World Almanac and Book of Facts) 1912 148.3 1920 146.6 1924 151.3 1928 155.3 1932 162.3 1936 165.6 1948 173.2 1952 180.5 1956 184.9 1960 194.2 1964 200.1 1968 212.5 1972 211.3 1976 221.5 1980 218.7 1984 218.5 1988 225.8 1992 213.7 1996 227.7 2000 227.3 2004 229.3 (a) Sketch a scatter plot of the data. Let represent the length of the winning discus throw (in feet) and let t 12 represent 1912. y (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the winning menโs discus throw in the year 2008. (f) Use your schoolโs library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e). 8. Revenue The total revenues (in millions of dollars) for Outback Steakhouse from 1995 to 2003 are listed below. (Source: Outback Steakhouse, Inc.) 1995 664.0 1996 937.4 1997 1151.6 1998 1358.9 1999 1646.0 2000 1906.0 2001 2127.0 2002 2362.1 2003 2744.4 (a) Sketch a scatter plot of the data. Let y revenue (in millions of dollars) and let 1995. represent the total represent t 5 (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the revenues |
of Outback Steakhouse in 2005. (f) Use your schoolโs library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e). 9. Data Analysis: Broadway Shows The table shows the S annual gross ticket sales (in millions of dollars) for Broadway shows in New York City from 1995 through 2004. (Source: The League of American Theatres and Producers, Inc.) Year Sales, S 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 406 436 499 558 588 603 666 643 721 771 (a) Use a graphing utility to create a scatter plot of the data. Let t 5 represent 1995. (b) Use the regression feature of a graphing utility to find the equation of the least squares regression line that fits the data. (c) Use the graphing utility to graph the scatter plot you found in part (a) and the model you found in part (b) in the same viewing window. How closely does the model represent the data? (d) Use the model to estimate the annual gross ticket sales in 2005 and 2007. (e) Interpret the meaning of the slope of the linear model in the context of the problem. x 10. Data Analysis: Television Households The table shows (in millions) of households with cable televithe numbers sion and the numbers (in millions) of households with color television sets in the United States from 1995 through 2002. (Source: Nielson Media Research; Television Bureau of Advertising, Inc.) y Households with cable, x Households with color TV, y 63 65 66 67 75 77 80 86 94 95 97 98 99 101 102 105 333202_0110.qxd 12/7/05 8:47 AM Page 111 Section 1.10 Mathematical Modeling and Variation 111 (a) Use the regression feature of a graphing utility to find the equation of the least squares regression line that fits the data. (b) Use the graphing utility to create a scatter plot of the data. Then graph the model you found in part (a) and the scatter plot in the same viewing window. How closely does the model represent the data? (c) Use the model to estimate the number of households with color television sets if the number of households with cable television is 90 million. (d) Interpret the meaning of the slope of the linear model in the context of the problem. Think About It In Exercises 11 and 12, use the |
graph to determine whether varies directly as some power of or inversely as some power of Explain. x. y x 11. y 12 In Exercises 13โ16, use the given value of k y kx2. table for the direct variation model on a rectangular coordinate system. to complete the Plot the points x 2 4 6 8 10 y kx2 13. 15. k 1 k 1 2 14. 16. k 2 k 1 4 In Exercises 17โ20, use the given value of k table for the inverse variation model to complete the y k x 2. Plot the points on a rectangular coordinate system. 2 4 6 8 10 x y k x2 17. 19. k 2 k 10 18. 20. k 5 k 20 In Exercises 21โ24, determine whether the variation model y k/x is of the form, and find y kx or k. 21. 22. 23. 24 10 15 20 25 1 2 1 3 1 4 1 5 10 15 20 4 6 8 25 10 5 3.5 10 7 15 20 25 10.5 14 17.5 5 10 15 20 25 24 12 8 6 24 5 Direct Variation In Exercises 25โ28, assume that x. directly proportional to y x -value to find a linear model that relates and is -value and x. Use the given y y 25. 26. 27. 28. x 5, x 2, x 10, x 6, y 12 y 14 y 2050 y 580 29. Simple Interest The simple interest on an investment is directly proportional to the amount of the investment. By investing $2500 in a certain bond issue, you obtained an interest payment of $87.50 after 1 year. Find a mathematical model that gives the interest for this bond issue after 1 year in terms of the amount invested P. I 30. Simple Interest The simple interest on an investment is directly proportional to the amount of the investment. By investing $5000 in a municipal bond, you obtained an interest payment of $187.50 after 1 year. Find a mathematfor this municipal bond ical model that gives the interest P. after 1 year in terms of the amount invested I 31. Measurement On a yardstick with scales in inches and centimeters, you notice that 13 inches is approximately the same length as 33 centimeters. Use this information to find a mathematical model that relates centimeters to inches. Then use the model to find the numbers of centimeters in 10 inches and 20 inches. 32. Measurement When buying |
gasoline, you notice that 14 gallons of gasoline is approximately the same amount of gasoline as 53 liters. Then use this information to find a linear model that relates gallons to liters. Then use the model to find the numbers of liters in 5 gallons and 25 gallons. 333202_0110.qxd 12/7/05 8:47 AM Page 112 112 Chapter 1 Functions and Their Graphs 33. Taxes Property tax is based on the assessed value of a property. A house that has an assessed value of $150,000 has a property tax of $5520. Find a mathematical model that gives the amount of property tax in terms of the assessed value of the property. Use the model to find the property tax on a house that has an assessed value of $200,000. y x 8 ft 34. Taxes State sales tax is based on retail price. An item that sells for $145.99 has a sales tax of $10.22. Find a in mathematical model that gives the amount of sales tax x. terms of the retail price Use the model to find the sales tax on a $540.50 purchase. y Hookeโs Law In Exercises 35โ38, use Hookeโs Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. 35. A force of 265 newtons stretches a spring 0.15 meter (see figure). FIGURE FOR 38 In Exercises 39โ48, find a mathematical model for the verbal statement. 39. A varies directly as the square of r. 40. V varies directly as the cube of e. 41. y varies inversely as the square of x. Equilibrium 0.15 meter 265 newtons (a) How far will a force of 90 newtons stretch the spring? (b) What force is required to stretch the spring 0.1 meter? 36. A force of 220 newtons stretches a spring 0.12 meter. What force is required to stretch the spring 0.16 meter? 37. The coiled spring of a toy supports the weight of a child. The spring is compressed a distance of 1.9 inches by the weight of a 25-pound child. The toy will not work properly if its spring is compressed more than 3 inches. What is the weight of the heaviest child who should be allowed to use the toy? 38. An overhead garage door has two springs, one on each side of the door (see |
figure). A force of 15 pounds is required to stretch each spring 1 foot. Because of a pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the combined lifting force applied to the door by the springs when the door is closed. 42. h varies inversely as the square root of 43. F g varies directly as and inversely as s. r 2. 44. z is jointly proportional to the square of and the cube of x y. 45. Boyleโs Law: For a constant temperature, the pressure of a gas is inversely proportional to the volume of the gas. V P 46. Newtonโs Law of Cooling: The rate of change R of the temperature of an object is proportional to the T difference between the temperature of the object and the of the environment in which the object is temperature placed. Te F 47. Newtonโs Law of Universal Gravitation: The gravitationis al attraction between two objects of masses proportional to the product of the masses and inversely proportional to the square of the distance between the objects. and m2 m1 r 48. Logistic Growth: The rate of growth of a population is jointly proportional to the size of the population and the L difference between and the maximum population size that the environment can support. R S S In Exercises 49โ54, write a sentence using the variation terminology of this section to describe the formula. 49. Area of a triangle: A 1 50. Surface area of a sphere: 2bh 51. Volume of a sphere: S 4r 2 r3 V 4 3 52. Volume of a right circular cylinder: V r 2h 53. Average speed: r d t 54. Free vibrations: kg W 333202_0110.qxd 12/7/05 8:47 AM Page 113 Section 1.10 Mathematical Modeling and Variation 113 In Exercises 55โ62, find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) when r 3. x 25. when x 4 and the third power of when y 7 when x 4. and y 8. s. 55. A 56. 57. 58. 59. 60. 61. 62. y y z x x. r 2. varies directly as varies jointly as and varies inversely as A 9 y 3 x. is inversely |
proportional to z 64 r s 3. is jointly proportional to and r 11 x varies directly as y 9. x 42 and when varies directly as the square of when y. 3 F F 4158 P P 28 z z 6 v v 1.5 x and inversely as y. and when y 4. q varies jointly as and and inversely as the square of q 6.3, x 6 p p 4.1, s 1.2. when and s. and inversely as the square of y. Ecology In Exercises 63 and 64, use the fact that the diameter of the largest particle that can be moved by a stream varies approximately directly as the square of the velocity of the stream. 63. A stream with a velocity of mile per hour can move coarse sand particles about 0.02 inch in diameter. Approximate the velocity required to carry particles 0.12 inch in diameter. 1 4 64. A stream of velocity can move particles of diameter or d increase when the velocity is d v less. By what factor does doubled? Resistance In Exercises 65 and 66, use the fact that the resistance of a wire carrying an electrical current is directly proportional to its length and inversely proportional to its cross-sectional area. 65. If #28 copper wire (which has a diameter of 0.0126 inch) has a resistance of 66.17 ohms per thousand feet, what length of #28 copper wire will produce a resistance of 33.5 ohms? 66. A 14-foot piece of copper wire produces a resistance of 0.05 ohm. Use the constant of proportionality from Exercise 65 to find the diameter of the wire. 67. Work The work W (in joules) done when lifting an object (in kilograms) of the object varies jointly with the mass and the height (in meters) that the object is lifted. The work done when a 120-kilogram object is lifted 1.8 meters is 2116.8 joules. How much work is done when lifting a 100-kilogram object 1.5 meters? m h 68. Spending The prices of three sizes of pizza at a pizza shop are as follows. 9-inch: $8.78, 12-inch: $11.78, 15-inch: $14.18 You would expect that the price of a certain size of pizza would be directly proportional to its surface area. Is that the case for this pizza shop? If not, which size of pizza is the |
best buy? 69. Fluid Flow The velocity of a fluid flowing in a conduit v is inversely proportional to the cross-sectional area of the conduit. (Assume that the volume of the flow per unit of time is held constant.) Determine the change in the velocity of water flowing from a hose when a person places a finger over the end of the hose to decrease its cross-sectional area by 25%. 70. Beam Load The maximum load that can be safely supported by a horizontal beam varies jointly as the width of the beam and the square of its depth, and inversely as the length of the beam. Determine the changes in the maximum safe load under the following conditions. (a) The width and length of the beam are doubled. (b) The width and depth of the beam are doubled. (c) All three of the dimensions are doubled. (d) The depth of the beam is halved. Model It 71. Data Analysis: Ocean Temperatures An oceanographer took readings of the water temperatures (in d (in meters). The degrees Celsius) at several depths data collected are shown in the table. C Depth, d Temperature, C 1000 2000 3000 4000 5000 4.2 1.9 1.4 1.2 0.9 (a) Sketch a scatter plot of the data. (b) Does it appear that the data can be modeled by the for C kd? If so, find k inverse variation model each pair of coordinates. k (c) Determine the mean value of C kd. the inverse variation model from part (b) to find (d) Use a graphing utility to plot the data points and the inverse model in part (c). (e) Use the model to approximate the depth at which the water temperature is C. 3 333202_0110.qxd 12/7/05 8:47 AM Page 114 114 Chapter 1 Functions and Their Graphs 72. Data Analysis: Physics Experiment An experiment in a physics lab requires a student to measure the compressed lengths (in centimeters) of a spring when various forces F of pounds are applied. The data are shown in the table. y Force, F Length, y 0 2 4 6 8 10 12 0 1.15 2.3 3.45 4.6 5.75 6.9 (a) Sketch a scatter plot of the data. (b) Does it appear that the data can be modeled by Hookeโs Law? If so, estimate k. (See Exerc |
ises 35โ38.) (c) Use the model in part (b) to approximate the force required to compress the spring 9 centimeters. 73. Data Analysis: Light Intensity A light probe is located x (in centimeters from a light source, and the intensity microwatts per square centimeter) of the light is measured. The results are shown as ordered pairs 30, 0.1881 42, 0.0998 A model for the data is x, y. 38, 0.1172 50, 0.0645 34, 0.1543 46, 0.0775 y 262.76x 2.12. y (a) Use a graphing utility to plot the data points and the model in the same viewing window. (b) Use the model to approximate the light intensity 25 centimeters from the light source. 74. Illumination The illumination from a light source varies inversely as the square of the distance from the light source. When the distance from a light source is doubled, how does the illumination change? Discuss this model in terms of the data given in Exercise 73. Give a possible explanation of the difference. Synthesis 78. Discuss how well the data shown in each scatter plot can be approximated by a linear model. (a) y (bcd 79. Writing A linear mathematical model for predicting prize winnings at a race is based on data for 3 years. Write a paragraph discussing the potential accuracy or inaccuracy of such a model. 80. Research Project Use your schoolโs library, the Internet, or some other reference source to find data that you think describe a linear relationship. Create a scatter plot of the data and find the least squares regression line that represents the data points. Interpret the slope and -intercept in the context of the data. Write a summary of your findings. y Skills Review In Exercises 81โ 84, solve the inequality and graph the solution on the real number line. 81. 82. 83. 3x 2 > 17 7x 10 โค 1 x 2x 1 < 9 84. 4 3x 7 โฅ 12 In Exercises 85 and 86, evaluate the function at each value of the independent variable and simplify. 85. f x x2 5 x 3 True or False? statement is true or false. Justify your answer. In Exercises 75โ77, decide whether the 75. If varies directly as y as well. x, then if x increases, will increase y 86. (a) f 0 (b) f |
3 (c) f 4 f x x2 10, 6x2 1, f 1 (b) f 2 (a) x โฅ 2 x < 2 (c) f 8 76. In the equation for kinetic energy, E 1 is directly proportional to the mass the amount m 2m v2, of kinetic energy of an object and the square of its velocity E v. 77. If the correlation coefficient for a least squares regression the regression line cannot be used to 1, line is close to describe the data. 87. Make a Decision To work an extended application analyzing registered voters in United States, visit this textโs website at college.hmco.com. (Data Source: U.S. Census Bureau) 333202_010R.qxd 12/7/05 8:49 AM Page 115 1 2 Chapter Summary What did you learn? Section 1.1 Plot points on the Cartesian plane (p. 2). Use the Distance Formula to find the distance between two points (p. 4). Use the Midpoint Formula to find the midpoint of a line segment (p. 5). Use a coordinate plane and geometric formulas to model and solve real-life problems (p. 6). Section 1.2 Sketch graphs of equations (p. 14). Find x- and y-intercepts of graphs of equations (p. 17). Use symmetry to sketch graphs of equations (p. 18). Find equations of and sketch graphs of circles (p. 20). Use graphs of equations in solving real-life problems (p. 21). Section 1.3 Use slope to graph linear equations in two variables (p. 25). Find slopes of lines (p. 27). Write linear equations in two variables (p. 29). Use slope to identify parallel and perpendicular lines (p. 30). Use slope and linear equations in two variables to model and solve real-life problems (p. 31). Section 1.4 Determine whether relations between two variables are functions (p. 40). Use function notation and evaluate functions (p. 42). Find the domains of functions (p. 44). Use functions to model and solve real-life problems (p. 45). Evaluate difference quotients (p. 46). Section 1.5 Use the Vertical Line Test for functions (p. 54). Find the zeros of functions (p. 56). Determine intervals on which functions are increasing or decreasing and determine relative maximum and relative minimum values of functions (p. 57). Determine the |
average rate of change of a function (p. 59). Identify even and odd functions (p. 60). Chapter Summary 115 Review Exercises 1โ4 5โ8 5โ8 9โ14 15โ24 25โ28 29โ36 37โ 44 45, 46 47โ50 51โ54 55โ62 63, 64 65, 66 67โ70 71, 72 73โ76 77, 78 79, 80 81โ84 85โ88 89โ94 95โ98 99โ102 333202_010R.qxd 12/7/05 8:49 AM Page 116 116 Chapter 1 Functions and Their Graphs Section 1.6 Identify and graph linear, squaring (p. 66), cubic, square root, reciprocal (p. 68), step, and other piecewise-defined functions (p. 69). Recognize graphs of parent functions (p. 70). Section 1.7 Use vertical and horizontal shifts to sketch graphs of functions (p. 74). Use reflections to sketch graphs of functions (p. 76). Use nonrigid transformations to sketch graphs of functions (p. 78). Section 1.8 Add, subtract, multiply, and divide functions (p. 84). Find the composition of one function with another function (p. 86). Use combinations and compositions of functions to model and solve real-life problems (p. 88). Section 1.9 Find inverse functions informally and verify that two functions are inverse functions of each other (p. 93). Use graphs of functions to determine whether functions have inverse functions (p. 95). Use the Horizontal Line Test to determine if functions are one-to-one (p. 96). Find inverse functions algebraically (p. 97). Section 1.10 Use mathematical models to approximate sets of data points (p. 103). Use the regression feature of a graphing utility to find the equation of a least squares regression line (p. 104). Write mathematical models for direct variation (p. 105). Write mathematical models for direct variation as an nth power (p. 106). Write mathematical models for inverse variation (p. 107). Write mathematical models for joint variation (p. 108). 103โ114 115, 116 117โ120 121โ126 127โ130 131, 132 133โ136 137, 138 139, 140 141, 142 143โ146 147โ152 153 154 155 156, 157 158, 159 160 333202_010R.qxd 12/7/05 8:49 AM Page 117 1 Review |
Exercises In Exercises 1 and 2, plot the points in the Cartesian 1.1 plane. 1. 2. 2, 2, 0, 4, 3, 6, 1, 7 5, 0, 8, 1, 4, 2, 3, 3 In Exercises 3 and 4, determine the quadrant(s) in which x, y is located so that the condition(s) is (are) satisfied. 3. x > 0 and y 2 4. y > 0 In Exercises 5โ8, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. (a) plot the points, 5. 6. 7. 8. 1, 5 4, 3 0, 8.2 3, 8, 2, 6, 5.6, 0, 0, 1.2, Review Exercises 117 13. Geometry The volume of a globe is about 47,712.94 cubic centimeters. Find the radius of the globe. 14. Geometry The volume of a rectangular package is 2304 cubic inches. The length of the package is 3 times its width, and the height is 1.5 times its width. (a) Draw a diagram that represents the problem. Label the height, width, and length accordingly. (b) Find the dimensions of the package. In Exercises 15โ18, complete a table of values. Use 1.2 the solution points to sketch the graph of the equation. 15. 16. 17. 18. y 3x 5 y 1 2x 2 y x2 3x y 2x2 x 9 3.6, 0 In Exercises 19โ24, sketch the graph by hand. In Exercises 9 and 10, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position. 9. Original coordinates of vertices: 4, 8, 6, 8, 4, 3, 6, 3 Shift: three units downward, two units to the left 10. Original coordinates of vertices: 0, 1, 3, 3, 0, 5, 3, 3 Shift: five units upward, four units to the left 11. Sales The Cheesecake Factory had annual sales of $539.1 million in 2001 and $773.8 million in 2003. Use the Midpoint Formula to estimate the sales in 2002. (Source: The |
Cheesecake Factory, Inc.) 12. Meteorology The apparent temperature is a measure of relative discomfort to a person from heat and high x (in humidity. The table shows the actual temperatures y degrees Fahrenheit) versus the apparent temperatures (in degrees Fahrenheit) for a relative humidity of 75%. x y 70 70 75 77 80 85 85 95 90 95 100 109 130 150 (a) Sketch a scatter plot of the data shown in the table. (b) Find the change in the apparent temperature when the actual temperature changes from 70F to 100F. 19. 20. 21. 22. 23. 24. y 2x 3 0 3x 2y 6 0 y 5 x y x 2 y 2x2 0 y x2 4x In Exercises 25โ28, find the - and -intercepts of the graph of the equation. x y 25. 26. 27. 28. y 2x 7 y x 1 3 y x 32 4 y x4 x2 In Exercises 29โ36, use the algebraic tests to check for symmetry with respect to both axes and the origin. Then sketch the graph of the equation. 32. 30. 31. 29. y 4x 1 y 5x 6 y 5 x2 y x2 10 y x3 3 y 6 x3 y x 5 35. 36. y x 9 33. 34. 333202_010R.qxd 12/7/05 8:49 AM Page 118 118 Chapter 1 Functions and Their Graphs In Exercises 37โ42, find the center and radius of the circle and sketch its graph. 37. 38. 39. 40. 41. 42. x2 y2 9 x2 y2 4 x 22 y2 16 x2 y 82 81 x 1 x 42 y 3 2 y 12 36 2 100 2 2 which the endpoints of a diameter are 43. Find the standard form of the equation of the circle for 4, 6. 44. Find the standard form of the equation of the circle for and which the endpoints of a diameter are 4, 10. 2, 3 0, 0 and 45. Physics The force x spring inches from its natural length (see figure) is F (in pounds) required to stretch a F 5 4 x, 0 โค x โค 20. Natural length x in. F (a) Use the model to complete the table. x 0 4 8 12 16 20 Force, F (b) Sketch a graph of the model. (c) Use the |
graph to estimate the force necessary to stretch the spring 10 inches. 46. Number of Stores The numbers of Target stores for the years 1994 to 2003 can be approximated by the model N N 3.69t2 939, 4 โค t โค 13 t where 1994. is the time (in years), with (Source: Target Corp.) (a) Sketch a graph of the model. t 4 corresponding to (b) Use the graph to estimate the year in which the number of stores was 1300. In Exercises 47โ50, find the slope and -intercept (if 1.3 possible) of the equation of the line. Sketch the line. y 47. 48. 49. 50. y 6 x 3 y 3x 13 y 10x 9 In Exercises 51โ54, plot the points and find the slope of the line passing through the pair of points. 51. 52. 53. 54. 7, 1 6, 5 3, 4, 1, 8, 4.5, 6, 2.1, 3 3, 2, 8, 2 In Exercises 55โ58, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope. Sketch the line. Point 0, 5 2, 6 10, 3 8, 5 55. 56. 57. 58. Slope m 3 2 m 0 m 1 2 m is undefined. In Exercises 59โ62, find the slope-intercept form of the equation of the line passing through the points. 59. 60. 61. 62. 2, 1 0, 0, 0, 10 2, 5, 1, 4, 2, 0 11, 2, 6, 1 In Exercises 63 and 64, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. Point 3, 2 8, 3 63. 64. Line 5x 4y 8 2x 3y 5 Rate of Change In Exercises 65 and 66, you are given the dollar value of a product in 2006 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value of the product in terms of the year represent 2006.) t 6 (Let V t. 2006 Value Rate 65. $12,500 66. $72.95 $850 increase |
per year $5.15 increase per year 333202_010R.qxd 12/7/05 8:49 AM Page 119 Review Exercises 119 In Exercises 67โ70, determine whether the equation 1.4 represents as a function of y x. 67. 68. 69. 70. 16x y 4 0 2x y 3 0 y 1 x y x 2 In Exercises 71 and 72, evaluate the function at each specified value of the independent variable and simplify. 81. y x 32 y 5 4 3 2 1 71. 72. (a) f x x 2 1 f 2 hx 2x 1, x2 2, h2 (a) (b) (b) f 4 x โค 1 x > 1 h1 (c) f t 2 (d) f t 1 โ1 1 2 3 4 5 83. x 4 y2 (c) h0 (d) h2 1.5 In Exercises 81โ84, use the Vertical Line Test to determine whether is a function of To print an enlarged copy of the graph, go to the website www.mathgraphs.com. x. y 82. y 3 5x 3 2x 1 y 1 โ3 โ2 โ1 1 2 3 โ2 โ3 84. x 4 y y 10 8 4 2 โ8 โ4 โ2 โ4 2 4 8 73. In Exercises 73โ76, find the domain of the function. Verify your result with a graph. f x 25 x 2 f x 3x 4 x x2 x 6 75. 74. h(x) h(t) t 1 76. 77. Physics The velocity of a ball projected upward from vt is the 32t 48, is the velocity in feet per second. where ground level is given by time in seconds and v t (a) Find the velocity when t 1. (b) Find the time when the ball reaches its maximum height. [Hint: Find the time when (c) Find the velocity when t 2. vt 0. ] 78. Mixture Problem From a full 50-liter container of a 40% liters is removed and replaced with x concentration of acid, 100% acid. (a) Write the amount of acid in the final mixture as a function of x. (b) Determine the domain and range of the function. (c) Determine x if the final mixture is 50% acid. In Exercises 79 and |
80, find the difference quotient and simplify your answer. 79. f x 2x2 3x 1, 80. f x x3 5x2 x In Exercises 85โ 88, find the zeros of the function algebraically. 85. 86. 87. fx 3x2 16x 21 fx 5x2 4x 1 f x 8x 3 11 x 88. fx x3 x 2 25x 25 In Exercises 89 and 90, determine the intervals over which the function is increasing, decreasing, or constant. f x x2 42 89. 90 20 8 4 โ2 โ1 21 3 x โ2 โ1 21 3 x 333202_010R.qxd 12/7/05 8:49 AM Page 120 120 Chapter 1 Functions and Their Graphs In Exercises 91โ94, use a graphing utility to graph the function and approximate (to two decimal places) any relative minimum or relative maximum values. 91. 92. 93. 94. fx x2 2x 1 fx x4 4x2 2 fx x3 6x4 fx x3 4x2 x 1 In Exercises 95โ98, find the average rate of change of the function from to x1 x2. Function fx x2 8x 4 fx x3 12x 2 fx 2 x 1 fx 1 x 3 95. 96. 97. 98. x -Values 0, x2 0, x2 3, x2 1, x2 4 4 7 6 x1 x1 x1 x1 In Exercises 99โ102, determine whether the function is even, odd, or neither. 99. 100. 101. 102. f x x 5 4x 7 f x x 4 20x2 f x 2xx2 3 f x 56x2 1.6 In Exercises 103โ104, write the linear function such that it has the indicated function values. Then sketch the graph of the function. f 2 6, f 0 5, f 1 3 f 4 8 104. 103. f In Exercises 105โ114, graph the function. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. f x 3 x2 hx x3 2 f x x f x x 1 gx 3 x gx 1 x 5 f x x 2 gx x 4 f x 5x 3, 4x 5, f x x2 2 |
, 5, 8x 5 In Exercises 115 and 116, the figure shows the graph of a transformed parent function. Identify the parent function. 115. y 116. y 10 8 6 4 2 8 6 4 2 โ8 โ4 โ2 2 x โ2 โ2 2 4 6 8 x f f. h. h. 117. 118. 121. 120. 119. In Exercises 117โ130, (c) Sketch the graph of in terms of 1.7 h is related to one of the parent functions described in this chapter. (a) Identify the parent f. (b) Describe the sequence of transformations function to from (d) Use function h notation to write hx x2 9 hx x 23 2 hx x 7 hx x 3 5 hx x 32 1 hx x 53 5 hx x 6 hx x 1 9 hx x 4 6 hx x 12 3 hx 5x 9 hx 1 hx 2x 4 2x 1 hx 1 126. 125. 128. 123. 127. 129. 130. 122. 124. 3 x 3 In Exercises 131 and 132, find (a) and (d) (b) What is the domain of fgx, (c) f gx, f/gx. 1.8 f gx, f/g? 131. 132. f x x2 3, f x x2 4, gx 2x 1 gx 3 x In Exercises 133 and 134, find (a) Find the domain of each function and each composite function. gx 3x 1 gx 3x 7 f x 1 3 x 3, f x x3 4, and (b) g f. 133. 134. f g In Exercises 135 and 136, find two functions and that (There are many correct answers.) f g such f gx hx. hx 6x 53 135. 136. hx 3x 2 333202_010R.qxd 12/7/05 8:49 AM Page 121 137. Electronics Sales The factory sales (in millions of from 1997 to vt dt dollars) for VCRs 2003 can be approximated by the functions vt 31.86t2 233.6t 2594 and DVD players and dt 4.18t2 571.0t 3706 t where 1997. represents the year, with (Source: Consumer Electronics Association) corresponding |
to t 7 (a) Find and interpret v dt. (b) Use a graphing utility to graph vt, dt, and the function from part (a) in the same viewing window. (c) Find v d10. Use the graph in part (b) to verify your result. 138. Bacteria Count The number N of bacteria in a refriger- ated food is given by NT 25T 2 50T 300, 2 โค T โค 20 T where is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by Tt 2t 1, 0 โค t โค 9 is the time in hours (a) Find the composition and interpret its meaning in context, and (b) find t where NTt, the time when the bacterial count reaches 750. In Exercises 139 and 140, find the inverse function of f f1x x and f 1fx x. 1.9 f informally. Verify that f x x 7 f x x 5 139. 140. In Exercises 141 and 142, determine whether the function has an inverse function. 141. y 4 2 โ2 2 4 x โ4 142. y x 2 4 โ2 โ2 โ4 โ6 In Exercises 143โ146, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. 143. 144. f x 4 1 3x f x x 12 Review Exercises 121 145. ht 2 t 3 146. gx x 6 f, In Exercises 147โ150, (a) find the inverse function of on the same set of coordinate axes, (b) graph both and (c) describe the relationship between the graphs of and f 1, and (d) state the domains and ranges of and f f 1. f 1 f f 147. 148. 149. 150. f x 1 2x 3 f x 5x 7 f x x 1 f x x3 2 In Exercises 151 and 152, restrict the domain of the function to an interval over which the function is increasing and determine over that interval. f 1 f 151. 152. f x 2x 42 f x x 2 1.10 I 153. Median Income The median incomes (in thousands of dollars) for married-couple families in the United States from 1995 through 2002 are shown in the table |
. A linear model that approximates these data is I 2.09t 37.2 t where 1995. t 5 represents the year, with (Source: U.S. Census Bureau) corresponding to Year 1995 1996 1997 1998 1999 2000 2001 2002 Median income, I 47.1 49.7 51.6 54.2 56.5 59.1 60.3 61.1 (a) Plot the actual data and the model on the same set of coordinate axes. (b) How closely does the model represent the data? 333202_010R.qxd 12/7/05 8:49 AM Page 122 122 Chapter 1 Functions and Their Graphs 154. Data Analysis: Electronic Games The table shows the factory sales (in millions of dollars) of electronic gaming software in the United States from 1995 through 2003. (Source: Consumer Electronics Association) S Year Sales, S 1995 1996 1997 1998 1999 2000 2001 2002 2003 3000 3500 3900 4480 5100 5850 6725 7375 7744 (a) Use a graphing utility to create a scatter plot of the data. corresponding represent the year, with t 5 t Let to 1995. (b) Use the regression feature of the graphing utility to find the equation of the least squares regression line that fits the data. Then graph the model and the scatter plot you found in part (a) in the same viewing window. How closely does the model represent the data? (c) Use the model to estimate the factory sales of electronic gaming software in the year 2008. (d) Interpret the meaning of the slope of the linear model in the context of the problem. 155. Measurement You notice a billboard indicating that it is 2.5 miles or 4 kilometers to the next restaurant of a national fast-food chain. Use this information to find a mathematical model that relates miles to kilometers. Then use the model to find the numbers of kilometers in 2 miles and 10 miles. 157. Frictional Force The frictional force between the tires and the road required to keep a car on a curved section of a highway is directly proportional to the square of the speed of the car. If the speed of the car is doubled, the force will change by what factor? F s 158. Demand A company has found that the daily demand x for its boxes of chocolates is inversely proportional to the price When the price is $5, the demand is 800 boxes. Approximate the demand when the price is increased to $6. p. 159. Travel |
Time The travel time between two cities is inversely proportional to the average speed. A train travels between the cities in 3 hours at an average speed of 65 miles per hour. How long would it take to travel between the cities at an average speed of 80 miles per hour? 160. Cost The cost of constructing a wooden box with a square base varies jointly as the height of the box and the square of the width of the box. A box of height 16 inches and width 6 inches costs $28.80. How much would a box of height 14 inches and width 8 inches cost? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 161โ163, determine whether 161. Relative to the graph of hx x 9 13 13 units downward, then reflected in the -axis. f x x, the function given by is shifted 9 units to the left and x f 162. If and are two inverse functions, then the domain of f. is equal to the range of g g 163. If y is directly proportional to x, then x is directly proportional to y. 164. Writing Explain the difference between the Vertical Line Test and the Horizontal Line Test. 165. Writing Explain how to tell whether a relation between two variables is a function. 156. Energy The power P produced by a wind turbine is proportional to the cube of the wind speed A wind speed of 27 miles per hour produces a power output of 750 kilowatts. Find the output for a wind speed of 40 miles per hour. S. 333202_010R.qxd 12/7/05 8:49 AM Page 123 1 Chapter Test Chapter Test 123 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Plot the points 6, 0. ment joining the points and the distance between the points. 2, 5 and Find the coordinates of the midpoint of the line seg- 2. A cylindrical can has a volume of 600 cubic centimeters and a radius of 4 centimeters. Find the height of the can. (โ3, 3) y 8 6 4 2 โ2 โ2 FIGURE FOR 6 In Exercises 3โ5, use intercepts and symmetry to sketch the graph of the equation. y 4 x y 3 5x y x2 1 4. 3. 5. (5, 3) x 4 6 6. Write the standard form of the equation of |
the circle shown at the left. In Exercises 7 and 8, find an equation of the line passing through the points. 7. 2, 3, 4, 9 8. 3, 0.8, 7, 6 9. Find equations of the lines that pass through the point 3, 8 and are (a) parallel to and (b) perpendicular to the line 4x 7y 5. 10. Evaluate x 9 x2 81 11. Determine the domain of f x at each value: (a) f 7 (b) f 5 (c) f x 9. f x 100 x2. In Exercises 12โ14, (a) find the zeros of the function, (b) use a graphing utility to graph the function, (c) approximate the intervals over which the function is increasing, decreasing, or constant, and (d) determine whether the function is even, odd, or neither. 12. f x 2x 6 5x 4 x2 13. f x 4x3 x 14. 15. Sketch the graph of f x 3x 7, 4x2 1 In Exercises 16 and 17, identify the parent function in the transformation. Then sketch a graph of the function. 16. hx x 17. hx x 5 8 In Exercises 18 and 19, find (a) and (f) (e) g f x. f gx, f gx, (b) f gx, (c) fgx, (d) f/gx, 18. f x 3x2 7, gx x2 4x 5 19. f x 1 x, gx 2x In Exercises 20โ22, determine whether or not the function has an inverse function, and if so, find the inverse function. 20. f x x 3 8 21. f x x2 3 6 22. f x 3xx In Exercises 23โ25, find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) v 24 when 23. s. v varies directly as the square root of A 500 varies jointly as and x 24. A s 16. y 8. y. b 32 when when a 1.5. x 15 and 25. b varies inversely as a. 333202_010R.qxd_pg 124 1/9/06 8:53 AM Page 124 Proofs in Mathematics What does the word proof mean to |
you? In mathematics, the word proof is used to mean simply a valid argument. When you are proving a statement or theorem, you must use facts, definitions, and accepted properties in a logical order. You can also use previously proved theorems in your proof. For instance, the Distance Formula is used in the proof of the Midpoint Formula below. There are several different proof methods, which you will see in later chapters. The Midpoint Formula The midpoint of the line segment joining the points given by the Midpoint Formula (p. 5) x1, y1 and x2, y2 is Midpoint x1 x2 2 y1, y2 2. Proof Using the figure, you must show that y (x1, y1) d1 d2 and d1 d2 d3. d1 d3 ( x1 + x 2 2, y1 + y2 2 ) d2 (x 2, y2) x By the Distance Formula, you obtain x1 d1 x2 2 x12 y1 y2 2 y12 1 2 x2 x1 2 y2 y1 2 x2 d2 x1 x2 2 2 y2 y1 y2 2 2 x1 2 y2 y1 2 x2 1 2 x2 d3 So, it follows that x1 d1 2 y2 d2 y1 and d1 2 d2 d3. The Cartesian Plane The Cartesian plane was named after the French mathematician Renรฉ Descartes (1596โ1650). While Descartes was lying in bed, he noticed a fly buzzing around on the square ceiling tiles. He discovered that the position of the fly could be described by which ceiling tile the fly landed on. This led to the development of the Cartesian plane. Descartes felt that a coordinate plane could be used to facilitate description of the positions of objects. 124 333202_010R.qxd 12/7/05 2:49 PM Page 125 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. As a salesperson, you receive a monthly salary of $2000, plus a commission of 7% of sales. You are offered a new job at $2300 per month, plus a commission of 5% of sales. (a) Write a linear equation for your current monthly wage W1 in terms of your monthly sales S. W2 (b) Write |
a linear equation for the monthly wage S. new job offer in terms of the monthly sales of your (c) Use a graphing utility to graph both equations in the same viewing window. Find the point of intersection. What does it signify? (d) You think you can sell $20,000 per month. Should you change jobs? Explain. 2. For the numbers 2 through 9 on a telephone keypad (see figure), create two relations: one mapping numbers onto letters, and the other mapping letters onto numbers. Are both relations functions? Explain. y (x, y) 12 ft FIGURE FOR 6 8 ft x 7. At 2:00 P.M. on April 11, 1912, the Titanic left Cobh, Ireland, on her voyage to New York City. At 11:40 P.M. on April 14, the Titanic struck an iceberg and sank, having covered only about 2100 miles of the approximately 3400-mile trip. (a) What was the total duration of the voyage in hours? (b) What was the average speed in miles per hour? (c) Write a function relating the distance of the Titantic from New York City and the number of hours traveled. Find the domain and range of the function. (d) Graph the function from part (c). 3. What can be said about the sum and difference of each of the following? (a) Two even functions (b) Two odd functions (c) An odd function and an even function 4. The two functions given by gx x f x x and are their own inverse functions. Graph each function and explain why this is true. Graph other linear functions that are their own inverse functions. Find a general formula for a family of linear functions that are their own inverse functions. 5. Prove that a function of the following form is even. y a2nx2n a2n2x2n2... a2x2 a0 6. A miniature golf professional is trying to make a hole-inone on the miniature golf green shown. A coordinate plane is placed over the golf green. The golf ball is at the point 2.5, 2 The professional wants to bank the ball off the side wall of the green at the point Then Find the coordinates of the point write an equation for the path of the ball. and the hole is at the point 9.5, 2. x, y. x, y. 8. Consider the function given by the average rate |
of change of the function from fx x2 4x 3. x2. to Find x1 1.5 (b) x1 1, x2 (a) (c) (d) (e) x1 x1 x1 x1 1, x2 1, x2 1, x2 1, x2 2 1.25 1.125 1.0625 (f) Does the average rate of change seem to be approaching one value? If so, what value? (g) Find the equations of the secant lines through the points x1, fx1 and x2, fx2 for parts (a)โ(e). (h) Find the equation of the line through the point 1, f1 using your answer from part (f ) as the slope of the line. gx x 6. f x 4x and (a) Find 9. Consider the functions given by f gx. f g1x. f 1x and g1 f 1x (b) Find (c) Find (d) Find g1x. and compare the result with that of part (b). (e) Repeat parts (a) through (d) for gx 2x. f x x3 1 and (f) Write two one-to-one functions and (a) through (d) for these functions. f g, and repeat parts (g) Make a conjecture about f g1x and g1 f 1x. 125 333202_010R.qxd 12/7/05 8:49 AM Page 126 10. You are in a boat 2 miles from the nearest point on the coast. You are to travel to a point 3 miles down the coast and 1 mile inland (see figure). You can row at 2 miles per hour and you can walk at 4 miles per hour. Q, 13. Show that the Associative Property holds for compositions of functionsโthat is, f g hx f g hx. 14. Consider the graph of the function shown in the figure. Use this graph to sketch the graph of each function. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. f (a) (e) f x 1 f x (b) (f) f x 1 f x (c) (g) 2f x f x (d) f x y 4 2 โ4 โ2 2 4 x โ2 โ4 15. Use the |
graphs of function values. f and f1 to complete each table of y 4 2 โ2 โ2 โ1 โ2 โ2 โ4 (a) x 4 2 0 4 f f 1x (b) x 3 2 0 1 (c) (d) f f 1x x f f 1x x f 1x 3 2 0 1 4 3 0 4 2 mi x 3 โ x 1 mi 3 mi Q Not drawn to scale. (a) Write the total time of the trip as a function of T x. (b) Determine the domain of the function. (c) Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. (d) Use the zoom and trace features to find the value of x that minimizes T. 11. The Heaviside function (e) Write a brief paragraph interpreting these values. Hx is widely used in engineering applications. (See figure.) To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Hx 1, 0, x โฅ 0 x < 0 Sketch the graph of each function by hand. Hx Hx 2 2 Hx 2 Hx (b) (d) (e) (c) (a) (f) Hx 2 2 Hx 1 y 3 2 1 โ3 โ2 โ1 1 2 3 x โ2 โ3 12. Let f x 1 1 x. (a) What are the domain and range of f? f f x. f f f x. What is the domain of this function? Is the graph a line? Why or why not? (b) Find (c) Find 126 22 333202_0200.qxd 12/7/05 9:08 AM Page 127 Polynomial and Rational Functions 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Quadratic Functions and Models Polynomial Functions of Higher Degree Polynomial and Synthetic Division Complex Numbers Zeros of Polynomial Functions Rational Functions Nonlinear Inequalities Quadratic functions are often used to model real-life phenomena, such as the path of a diver AT I O N S Polynomial and rational functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. โข Path of a Diver, Exercise 77, page 136 โข Data Analysis: Home Prices, Exercises 93โ96, page 151 โข |
Advertising Cost, Exercise 105, page 181 โข Athletics, Exercise 109, page 182 โข Data Analysis: Cable Television, โข Recycling, Exercise 74, page 161 Exercise 112, page 195 โข Average Speed, Exercise 79, page 196 โข Height of a Projectile, Exercise 67, page 205 127 333202_0201.qxd 12/7/05 9:10 AM Page 128 128 Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions and Models What you should learn โข Analyze graphs of quadratic functions. โข Write quadratic functions in standard form and use the results to sketch graphs of functions. โข Use quadratic functions to model and solve real-life problems. Why you should learn it Quadratic functions can be used to model data to analyze consumer behavior. For instance, in Exercise 83 on page 137, you will use a quadratic function to model the revenue earned from manufacturing handheld video games. ยฉ John Henley/Corbis The Graph of a Quadratic Function In this and the next section, you will study the graphs of polynomial functions. In Section 1.6, you were introduced to the following basic functions. f x ax b f x c f x x2 Linear function Constant function Squaring function These functions are examples of polynomial functions. Definition of Polynomial Function Let be a nonnegative integer and let 0. numbers with an, n an The function given by f x anx n an1x n1... a 2x 2 a1x a 0 an1,..., a2, a1, a0 be real is called a polynomial function of x with degree n. Polynomial functions are classified by degree. For instance, a constant function has degree 0 and a linear function has degree 1. In this section, you will study second-degree polynomial functions, which are called quadratic functions. For instance, each of the following functions is a quadratic function. f x x2 6x 2 gx 2x 12 3 hx 9 1 kx 3x2 4 mx x 2x 1 4 x2 Note that the squaring function is a simple quadratic function that has degree 2. Definition of Quadratic Function a 0. and be real numbers with Let c a, b, f x ax 2 bx c Quadratic function The function given by is called a quadratic function. The HM mathSpaceยฎ CD- |
ROM and Eduspaceยฎ for this text contain additional resources related to the concepts discussed in this chapter. The graph of a quadratic function is a special type of โUโ-shaped curve called a parabola. Parabolas occur in many real-life applicationsโespecially those involving reflective properties of satellite dishes and flashlight reflectors. You will study these properties in Section 10.2. 333202_0201.qxd 12/7/05 9:10 AM Page 129 Section 2.1 Quadratic Functions and Models 129 All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola, as shown in Figure 2.1. If the leading coefficient is positive, the graph of f x ax 2 bx c is a parabola that opens upward. If the leading coefficient is negative, the graph of f x ax 2 bx c is a parabola that opens downward. y y Opens upward Axis f x ( ) = ax 2 + bx + c, a < 0 Vertex is highest point Axis Vertex is lowest point f x ( ) = ax 2 + bx + c, a 0 > x Leading coefficient is positive. FIGURE 2.1 x Opens downward Leading coefficient is negative. The simplest type of quadratic function is f x ax 2. Its graph is a parabola whose vertex is (0, 0). If the minimum -value on the graph, and if maximum -value on the graph, as shown in Figure 2.2. a < 0, a > 0, y y the vertex is the point with the vertex is the point with the y 3 2 1 โ1 โ2 โ3 โ3 โ2 โ1 f x ( ) = 2 ax a, > 0 x 1 2 3 Minimum: (0, 0) y 3 2 1 Maximum: (0, 0) โ3 โ2 โ ax a, < 0 โ1 โ2 โ3 Leading coefficient is positive. FIGURE 2.2 Leading coefficient is negative. When sketching the graph of f x ax 2, it is helpful to use the graph of y x 2 as a reference, as discussed in Section 1.7. Exploration y ax2 Graph for 0.5, 0.5, 1, and 2. How does changing the value of affect the graph? |
a 2, 1, a for y x h2 h 4, 2, and 4. How does chang- Graph 2, ing the value of affect the graph? h for y x2 k k 4, 2, and 4. How does chang- Graph 2, ing the value of affect the graph? k 333202_0201.qxd 12/7/05 9:10 AM Page 130 130 Chapter 2 Polynomial and Rational Functions Example 1 Sketching Graphs of Quadratic Functions a. Compare the graphs of b. Compare the graphs of y x2 y x2 and and f x 1 3x2. gx 2x2. Solution a. Compared with 3x 2 creating the broader parabola shown in Figure 2.3. each output of y x 2, f x 1 โshrinksโ by a factor of 1 3, b. Compared with y x 2, each output of gx 2x 2 โstretchesโ by a factor of 2, creating the narrower parabola shown in Figure 2.4. y x2 โ1 FIGURE 2.3 x 1 2 โ2 โ1 FIGURE 2.4 Now try Exercise 9. y x= 2 1 2 x a In Example 1, note that the coefficient determines how widely the parabola is small, the parabola opens more widely than f x ax 2 opens. If a given by a if is large. Recall from Section 1.7 that the graphs of and y f x y f x, For instance, in Figure 2.5, notice how the graph of to produce the graphs of f x x 2 1 and y f x ยฑ c, y f x ยฑ c, y f x. can be transformed y x 2 gx x 22 3. are rigid transformations of the graph of y 2 (0, 1) y x= 2 โ2 f(x) = โ x2 + 1 x 2 โ1 โ2 Reflection in x-axis followed by an upward shift of one unit FIGURE 2.5 g(x) = (x + 2)4 โ3 โ1 1 2 x โ2 โ3 (โ2, โ3) Left shift of two units followed by a downward shift of three units 333202_0201.qxd 12/7/05 9:10 AM Page 131 Section 2.1 Quadratic Functions and Models 131 The Standard Form of a Quadratic Function The standard form of a quadr |
atic function identifies four basic transformations of the graph of y x2. a. The factor a produces a vertical stretch or shrink. b. If a < 0, x in the -axis. the graph is reflected c. The factor x h2 a horizontal shift of units. represents h d. The term represents a k vertical shift of units. k f x ( ) = 2( + 2= 2 2 x 1 โ1 x = 2โ โ3 โ โ ( 2, 1) FIGURE 2.6 The standard form of a quadratic function is This form is especially convenient for sketching a parabola because it identifies the vertex of the parabola as f x ax h 2 k. h, k. Standard Form of a Quadratic Function The quadratic function given by f x ax h 2 k, a 0 is in standard form. The graph of x h line upward, and if and whose vertex is the point a < 0, f the parabola opens downward. is a parabola whose axis is the vertical the parabola opens h, k. a > 0, If To graph a parabola, it is helpful to begin by writing the quadratic function in standard form using the process of completing the square, as illustrated in Example 2. In this example, notice that when completing the square, you add and subtract the square of half the coefficient of within the parentheses instead of adding the value to each side of the equation as is done in Appendix A.5. x Example 2 Graphing a Parabola in Standard Form Sketch the graph of the parabola. f x 2x 2 8x 7 and identify the vertex and the axis of Solution Begin by writing the quadratic function in standard form. Notice that the first step in completing the square is to factor out any coefficient of that is not 1. x2 f x 2x 2 8x 7 2x 2 4x 7 2x 2 4x 4 4 7 Write original function. Factor 2 out of -terms. x Add and subtract 4 within parentheses. 422 After adding and subtracting 4 within the parentheses, you must now regroup the can be removed from inside the terms to form a perfect square trinomial. The parentheses; however, because of the 2 outside of the parentheses, you must multiply by 2, as shown below. 4 4 f x 2x 2 4x 4 24 7 2x 2 4x 4 8 7 2x 22 1 Regroup terms. |
Simplify. Write in standard form. is a parabola that opens From this form, you can see that the graph of This corresponds to a left shift of two upward and has its vertex at units and a downward shift of one unit relative to the graph of as shown in Figure 2.6. In the figure, you can see that the axis of the parabola is the vertical line through the vertex, 2, 1. x 2. y 2x 2, f Now try Exercise 13. 333202_0201.qxd 12/7/05 9:10 AM Page 132 132 Chapter 2 Polynomial and Rational Functions To find the -intercepts of the graph of x you must solve the equation If does not factor, you can x use the Quadratic Formula to find the -intercepts. Remember, however, that a parabola may not have -intercepts. ax2 bx c 0. ax2 bx c f x ax2 bx c, x Example 3 Finding the Vertex and x-Intercepts of a Parabola Sketch the graph of f x x 2 6x 8 and identify the vertex and -intercepts. x Solution f x x 2 6x 8 x 2 6x 8 x 2 6x 9 9 8 622 x 2 6x 9 9 8 x 32 1 Write original function. Factor 1 x out of -terms. Add and subtract 9 within parentheses. Regroup terms. Write in standard form. f is a parabola that opens downward with vertex From this form, you can see that 3, 1. x The -intercepts of the graph are determined as follows. x2 6x 8 0 x 2x Factor out Factor. 1. Set 1st factor equal to 0. Set 2nd factor equal to 0. So, the -intercepts are x 2, 0 and 4, 0, as shown in Figure 2.7. Now try Exercise 19. Example 4 Writing the Equation of a Parabola Write the standard form of the equation of the parabola whose vertex is that passes through the point as shown in Figure 2.8. 0, 0, 1, 2 and Solution Because the vertex of the parabola is at h, k 1, 2, the equation has the form f x ax 12 2. k Substitute for and h Because the parabola passes through the point 0, 0, it follows that in standard form. f 0 0. So |
, f(x) = โ(x โ 3)2 + 1 (3, 1) (2, 0) (4, 0) 1 3 5 x y = โx2 (1, 2) y = f(x) โ1 y 2 1 โ1 โ2 โ3 โ4 FIGURE 2.7 y 2 1 (0, 0) 1 x 0 a0 12 2 a 2 Substitute 0 for x; solve for a. which implies that the equation in standard form is f x 2x 12 2. FIGURE 2.8 Now try Exercise 43. 333202_0201.qxd 12/7/05 9:10 AM Page 133 Section 2.1 Quadratic Functions and Models 133 Applications Many applications involve finding the maximum or minimum value of a quadratic function. You can find the maximum or minimum value of a quadratic function by locating the vertex of the graph of the function. Vertex of a Parabola The vertex of the graph of f x ax2 bx c is b 2a, f b 2a. 1. If a > 0, 2. If a < 0, has a minimum at x b 2a has a maximum at x b 2a.. Example 5 The Maximum Height of a Baseball A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per with respect to the ground. The path of the baseball second and at an angle of is the height of is given by the function x the baseball (in feet) and is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball? 45 f x 0.0032x 2 x 3, where f x Solution From the given function, you can see that function has a maximum when reaches its maximum height when it is x b2a, and a 0.0032 b 1. Because the you can conclude that the baseball is feet from home plate, where x x x b 2a x b 2a 1 20.0032 156.25 feet. At this distance, 156.25 3 81.125 the maximum height is f 156.25 0.0032156.25 2 feet. The path of the baseball is shown in Figure 2.9. Now try Exercise 77. Example 6 Minimizing Cost A small local soft-drink manufacturer has daily production costs of C 70,000 120x 0.075x2, C is the number of units produced. How many |
units should be produced each day to yield a minimum cost? is the total cost (in dollars) and where x Baseball f(x) = โ0.0032x 2 + x + 3 (156.25, 81.125) x 200 100 Distance (in feet) 300 y 100 ) 80 60 40 20 FIGURE 2.9 Solution Use the fact that the function has a minimum when function you can see that 120 2(0.075 x b 2a a 0.075 800 units and b 120. x b2a. From the given So, producing each day will yield a minimum cost. Now try Exercise 83. 333202_0201.qxd 12/7/05 9:10 AM Page 134 134 Chapter 2 Polynomial and Rational Functions 2.1 Exercises The HM mathSpaceยฎ CD-ROM and Eduspaceยฎ for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. A polynomial function of degree and leading coefficient n f x anxn an1xn1... a1x a0 an a1 2. A ________ function is a second-degree polynomial function, and its graph is called a ________. is a ________ ________ and 0 an where is a function of the form n are ________ numbers. 3. The graph of a quadratic function is symmetric about its ________. 4. If the graph of a quadratic function opens upward, then its leading coefficient is ________ and the vertex of the graph is a ________. 5. If the graph of a quadratic function opens downward, then its leading coefficient is ________ and the vertex of the graph is a ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1โ 8, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] (a) (c) y 6 4 2 (b) y 6 4 2 โ4 (โ1, โ2) (โ 4, 0) โ 6 โ 4 โ2 2 y 6 4 2 โ2 (e) y 2 โ2 โ4 |
โ6 (g) y 2 4 6 (3, 2)โ 6 4 2 (2, 0) โ4 โ2 (0, โ2) 2 4 (4, 0) 2 4 6 8 (d) y โ2 โ4 โ6 (2, 4) x 6 (f) y 4 2 โ2 (h) 2 y 4 (0, 3) x 4 โ4 โ2 โ4 1. 3. 5. 7. f x x 22 f x x 2 2 f x 4 (x 2)2 f x x 32 2 2. 4. 6. 8. f x x 42 f x 3 x 2 f x x 12 2 f x x 42 In Exercises 9โ12, graph each function. Compare the graph of each function with the graph of y x2. (b) (d) (b) (d) (b) (d) gx 1 8 x2 kx 3x 2 gx x 2 1 kx x 2 3 gx 3x2 1 kx x 32 9. (a) (c) 10. (a) (c) 11. (a) (c) 12. (a) (b) (c) (d) f x 1 2 x2 hx 3 2 x2 f x x 2 1 hx x 2 3 f x x 12 hx 1 3 x2 3 f x 1 x 22 1 2 gx 1 x 12 3 2 x 22 1 hx 1 2 kx 2x 12 4 In Exercises 13โ28, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and -intercept(s). x 14. 16. 18. 20. 22. 24. hx 25 x 2 f x 16 1 4 x 2 f x x 62 3 gx x 2 2x 1 f x x 2 3x 1 4 f x x 2 4x 1 13. 15. 17. 19. 21. 23. 25. 26. 27. 28. f x x 2 5 f x 1 2x 2 4 f x x 52 6 hx x 2 8x 16 2x 5 hx 4x 2 4x 21 f x 2x 2 x 1 f x 1 f x 1 4x2 2x 12 3x2 3x 6 333202_0201.qxd 12/7/05 9:10 AM Page 135 Section 2. |
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