Split (1)

train · 15k rows

text stringlengths 235 3.08k
, given each function f, evaluate f (−1), f (0), f (2), and f (4). { 49. f (x) = 7x + 3 if x < 0 7x + 6 if x ≥ 0 { 50. f (x) = if \| if x ≥ 2 51. f (x) = { 5x if 3 x 2 if x < 0 if 0 ≤ x ≤ 3 x > 3 For the following exercises, write the domain for the piecewise function in interval notation. { 52. f (x) = x + 1 if x < −2 ...
leARnInG OBjeCTIVeS In this section, you will: • • • • Find the average rate of change of a function. Use a graph to determine where a function is increasing, decreasing, or constant. Use a graph to locate local maxima and local minima. Use a graph to locate the absolute maximum and absolute minimum. 1. 3 RATeS OF CHA...
. It does not mean we are changing the function into some other function. In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was ∆y _ ∆x = $1.37 _ 7 years ≈ 0.196 dollars per year On average, the price of gas increased by about 19.6 each year. Other exampl...
= = Analysis Note that a decrease is expressed by a negative change or “negative increase.” A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases. = −$0.22 per year Try It #1 Using the data in Table 1 at the beginning of this section, find the...
282 ___ 6 = 47 Analysis Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2, 3], the average speed is 63 miles per hour. Computing Average Rate of Change for a Function Expressed as a Formula Example 4 Compute the average rate of change of f (x) = x 2 − 1 __ on the ...
) ___ a − 0 Simplify. = a2 + 3a + 1 − 1 _____________ a a(a + 3) _______ a = a + 3 = Simplify and factor. Divide by the common factor a. This result tells us the average rate of change in terms of a between t = 0 and any other point t = a. For example, on the interval [0, 5], the average rate of change would be 5 + 3 =...
singular form is “extremum.”) Often, the term local is replaced by the term relative. In this text, we will use the term local. Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of lo...
(x) for every point x (x does not equal both) in the interval. Example 7 Finding Increasing and Decreasing Intervals on a Graph Given the function p(t) in Figure 6, identify the intervals on which the function appears to be increasing. p 5 4 3 2 1 –1 1 2 3 4 5 6 t –1 –2 Figure 6 Solution We see that the function is no...
be symmetric, the two different technologies agree only up to four decimals due to the differing — approximation algorithms used by each. (The exact location of the extrema is at ± √ 6, but determining this requires calculus.) Try It #4 Graph the function f (x) = x3 − 6x2 − 15x + 20 to estimate the local extrema of th...
y Absolute Value f(x) = ∣ x ∣ Increasing on (0, ∞) Decreasing on (−∞, 0) Figure 12 x x x x x x SECTION 1.3 rates oF change and Behavior oF graphs 47 Use A Graph to locate the Absolute Maximum and Absolute Minimum There is a difference between locating the highest and lowest points on a graph in a region around an open...
practice with rates of change. • Average Rate of Change (http://openstaxcollege.org/l/aroc) 48 CHAPTER 1 Functions 1.3 SeCTIOn exeRCISeS VeRBAl 1. Can the average rate of change of a function be constant? 3. How are the absolute maximum and minimum similar to and different from the local extrema? 2. If a function f is...
–1 –2 –3 –4 –5 19. 21 3 4 5 x –5 –4 –3 –2 y 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5 20. 21 3 4 5 x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 x SECTION 1.3 section exercises 49 21. –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 21 3 4 5 6 x For the following exercises, consider the graph shown in Figure 16. 22. Estim...
. q(x) = x3 on [−4, 2] 1 _ x on [1, 3] 32. y = 4 __ t3 on [−1, 3] 34. k (t) = 6t2 + 31. g (x) = 3x 3 − 1 on [−3, 3] 33. p(t) = (t 2 − 4)(t + 1) ____________ t 2 + 3 on [−3, 1] TeCHnOlOGY For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on w...
to fill up his gas tank. He looked at his watch, and the time read exactly 3:40 p.m. At this time, he started pumping gas into the tank. At exactly 3:44, the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank? 46. Near the surface of the moo...
write C(T(5)). Cost for the temperature C(T(5)) Temperature on day 5 By combining these two relationships into one function, we have performed function composition, which is the focus of this section. Combining Functions Using Algebraic Operations Function composition is only one way to combine existing functions. Ano...
2 − 1. Are they the same Find and simplify the functions (g − f )(x) and  f function? Solution Begin by writing the general form, and then substitute the given functions. (g − f )(x) = g(x) − f (x) (g − f )(x) = x 2 − 1 − (x − 1) (g − f )(x) = x 2 − x (g − f )(x) = x(x − 1) x2 − 1 _ x − 1 g(x) _ f (x) g _  (x) =  f...
wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse ...
f. It is important to realize that the product of functions fg is not the same as the function composition f (g(x)), because, in general, f (x)g(x) ≠ f (g(x)). Example 2 Determining whether Composition of Functions is Commutative Using the functions provided, find f (g(x)) and g(f (x)). Determine whether the compositi...
: number of gallons = g (number of miles) The expression g(y) takes miles as the input and a number of gallons as the output. The function f (x) requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression f (g(y)) is meaningless. The expression f (x) takes hours as i...
1 g(x) 3 5 2 7 Solution To evaluate f (g(3)), we start from the inside with the input value 3. We then evaluate the inside expression g(3) using the table that defines the function g: g(3) = 2. We can then use that result as the input to the function f, so g(3) is replaced by 2 and we get f (2). Then, using the table ...
. g(x) f (x) 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 (1, 3) 321 4 5 6 7 x g(1) = 3 Figure 2 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 (3, 6) 321 4 5 6 7 x f (3) = 6 We evaluate g(1) using the graph of g(x), finding the input of 1 on the x-axis and finding the output value of the graph at that input. Here, g(1) = 3. We use this value as the in...
as the input to the outside function. Example 7 Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input Given f (t) = t2 − t and h(x) = 3x + 2, evaluate f (h(1)). Solution Because the inside expression is h(1), we start by evaluating h(x) at 1. Then f (h(1)) = f (5), so we evaluate f (t) at ...
… Given a function composition f (g(x)), determine its domain. 1. Find the domain of g. 2. Find the domain of f. 3. Find those inputs x in the domain of g for which g(x) is in the domain of f. That is, exclude those inputs x from the domain of g for which g(x) is not in the domain of f. The resulting set is the domain ...
range of functions (specifically the inner function) can also be helpful in finding the domain of a composite function. It also shows that the domain of f ∘ g can contain values that are not in the domain of f, though they must be in the domain of g. Try It #6 Find the domain of (f ∘ g)(x) where f (x) = 1 ____ x − 2 a...
f _ g? two functions, 3. If the order is reversed when composing two 4. How do you find the domain for the composition of functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not. two functions, f ∘ g? AlGeBRAIC 5. Given f (x) = ...
� — x + 2, g(x) = x 2 + 3 14. f (x) = \|x\|, g(x) = 5x + 1 15. f (x) = 3 √ 16. f (x(x) = — x, g(x) = x + 1 _ x3, g(x) = 2 _ x + 4 1 ___ x−4 17. f (x) = For the following exercises, use each set of functions to find f (g(h(x))). Simplify your answers. 18. f (x) = x 4 + 6, g(x) = x − 6, and h(x) = √ — x 1 _ x, and h(x) = x...
)(x) in interval notation. interval notation. a. q(x) _ h(x) b. q(h(x)) c. h(q(x)) For the following exercises, find functions f (x) and g(x) so the given function can be expressed as h(x) = f (g(x)). 26. h(x) = (x + 2)2 27. h(x) = (x − 5)3 30. h(x) = 4 + 3 √ — x 3 √ 31. h(x) = _______ 1 ______ 2x − 3 28. h(x) = 3 _ x ...
(x) h(x) f (x1 1 2 3 4 x x x Figure 6 Figure 7 Figure 8 50. g( f (1)) 51. g( f (2)) 52. f ( g(4)) 53. f ( g(1)) 54. f (h(2)) 55. h( f (2)) 56. f ( g(h(4))) 57. f ( g( f (−2))) nUMeRIC For the following exercises, use the function values for f and g shown in Table 3 to evaluate each expression. x f (x) g(x Table 58. f (...
f (x) = x3 + 1 and g(x) = 3 √ — x − 1. 80. Find ( f ∘ g)(x) and ( g ∘ f )(x). Compare the two answers. 81. Find ( f ∘ g)(2) and ( g ∘ f )(2). 82. What is the domain of ( g ∘ f )(x)? 83. What is the domain of ( f ∘ g)(x)? 1 __ 84. Let f (x) =. x a. Find ( f ∘ f )(x). b. Is ( f ∘ f )(x) for any function f the same resul...
Hint: Use function composition to find your answer.) 94. A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula r(t) = 2t + 1, express the area burned as a function of time, t (minutes). 96. The ...
• • • • • Combine transformations. Graph functions using vertical and horizontal shifts. Graph functions using reflections about the x-axis and the y-axis. Determine whether a function is even, odd, or neither from its graph. Graph functions using compressions and stretches. 1. 5 TRAnSFORMATIOn OF FUnCTIOnS Figure 1 (...
by y + k, so the y-value increases or decreases depending on the value of k. The result is a shift upward or downward. vertical shift Given a function f (x), a new function g(x) = f (x) + k, where k is a constant, is a vertical shift of the function f (x). All the output values change by k units. If k is positive, the...
ifting a Tabular Function Vertically A function f (x) is given in Table 2. Create a table for the function g(x) = f (x) − 3. x f (x) 2 1 4 3 Table 2 6 7 8 11 Solution The formula g (x) = f (x) − 3 tells us that we can find the output values of g by subtracting 3 from the output values of f. For example: Given Given tra...
the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in f. horizontal shift Given a function f, a new function g(x) = f (x − h), where h is a constant, is a horizontal shift of the function f. If h is positive, the graph will sh...
) = V(t + 2). How To… Given a tabular function, create a new row to represent a horizontal shift. 1. Identify the input row or column. 2. Determine the magnitude of the shift. 3. Add the shift to the value in each input cell. Example 4 Shifting a Tabular Function Horizontally A function f (x) is given in Table 4. Creat...
1 2 3 4 5 6 7 x Figure 8 Solution Notice that the graph is identical in shape to the f (x) = x2 function, but the x-values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so g (x) = f (x − 2) Notic...
+ 2) on the same axes. Combining Vertical and Horizontal Shifts Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output ( y-) axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input (x-) axis values...
y = \|x + 1\| – 3 21 3 4 5 x Figure 10 Try It #3 Given f (x) = ∣ x ∣, sketch a graph of h(x) = f (x − 2) + 4. SECTION 1.5 transFormation oF Functions 71 Example 8 Identifying Combined Vertical and Horizontal Shifts Write a formula for the graph shown in Figure 11, which is a transformation of the toolkit square root fun...
reflect the graph both vertically and horizontally. 1. Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the x-axis. 2. Multiply all inputs by –1 for a horizontal reflection. The new graph is a reflection of the original graph about the y-axis. Example 9 Re...
reflection gives the H(t) function the domain (−∞, 0]. Try It #5 Reflect the graph of f (x) = \|x − 1\| a. vertically and b. horizontally. Example 10 Reflecting a Tabular Function Horizontally and Vertically A function f (x) is given as Table 6. Create a table for the functions below. a. g(x) = −f (x) b. h(x) = f (−x) x...
of the three transformations. We will choose the points (0, 1) and (1, 2). 1. First, we apply a horizontal reflection: (0, 1) (−1, 2). 2. Then, we apply a vertical reflection: (0, −1) (1, −2). 3. Finally, we apply a vertical shift: (0, 0) (1, 1). This means that the original points, (0,1) and (1,2) become (0,0) and (1...
x5 –5 –4 –4 –3 –3 –2 –2 –1 –1 –1 –2 –3 –4 –5 –1 –2 –3 –4 –5 –5 –5 –4 –4 –3 –3 –2 –2 –1 –1 –1 –2 –3 –4 –5 –1 –2 –3 –4 –5 21 21 3 3 4 4 5 5 x x –5 –5 –4 –4 –3 –3 –2 –2 –1 –1 –1 –2 –3 –4 –5 –1 –2 –3 –4 –5 21 21 3 3 4 4 5 5 x x (a) (a) (b) (b) (c) (c) Figure 17 (a) The cubic toolkit function (b) Horizontal reflection of th...
us to the original function. Let’s begin with the rule for even functions. f (−x) = (−x)3 + 2(−x) = −x 3 − 2x This does not return us to the original function, so this function is not even. We can now test the rule for odd functions. Because −f (−x) = f (x), this is an odd function. −f (−x) = −(−x 3 − 2x) = x 3 + 2x 7...
constant, is a vertical stretch or vertical compression of the function f (x). • If a > 1, then the graph will be stretched. • If 0 < a < 1, then the graph will be compressed. • If a < 0, then there will be combination of a vertical stretch or compression with a vertical reflection. SECTION 1.5 transFormation oF Funct...
ular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression. 1. Determine the value of a. 2. Multiply all of the output values by a. Example 14 Finding a Vertical Compression of a Tabular Function A function f is given as Table 10. Create a table fo...
using the definition of the function f. f (x) = 1 g(x) = 1 __ __ x3 4 4 Try It #10 Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units. Horizontal Stretches and Compressions Now we consider changes to the inside of a function....
1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population. Solution Symbolically, we could write R(1) = P(2), R(2) = P(4), and in general, R(t) = P(2t). See Figure 24 for a graphical comparis...
–8 –10 –12 (b) 84 12 16 20 x Analysis Because each input value has been doubled, the result is that the function g(x) has been stretched horizontally by a factor of 2. Example 18 Recognizing a Horizontal Compression on a Graph Relate the function g(x) to f (x) in Figure 26. f (x) 6 5 4 3 2 1 – –1 1 Figure 26 Solution ...
example, we have to think about how the inputs to the function g relate to the inputs to the function f. Suppose we know f (7) = 12. What input to g would produce that output? In other words, what value of x will allow g(x) = f (2x + 3) = 12? We would need 2x + 3 = 7. To solve for x, we would first subtract 3, resulti...
f (3x) 2 20 4 28 Table 17 6 30 8 34 Finally, we can apply the vertical shift, which will add 1 to all the output values. See Table 18. x g(x) = 2f (3x) + 1 4 29 6 31 8 35 2 21 Table 18 Example 20 Finding a Triple Transformation of a Graph 1 __ Use the graph of f (x) in Figure 27 to sketch a graph of k(x. 2 f (x) –5 –4 ...
multiple transformations, how can you tell a horizontal compression from a vertical compression? 5. How can you determine whether a function is odd or even from the formula of the function? AlGeBRAIC the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch? 4. When examinin...
) = 2x − 3 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 –4 –3 –2 Figure 31 For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions. 27. f (t) = (t + 1)2 − 3 28. h(x) = \|x − 1\| + 4 29. k(x) = (x − 2)3 − 1 30. m(t) = 3 + √ — t + 2 86 nUMeRIC CHAPTER 1 Functions 31....
–4 –5 SECTION 1.5 section exercises 87 39. –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 40. f 21 3 4 5 6 x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 f 21 3 4 5 x For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions. 41. y 5 4 3 2 1 ...
x) 58. g(x) = f (2x) 61. g(x) = 3f (−x) 62. g(x) = −f (3x) 1 __ x  59. g(x) = f  3 1 __ x  60. g(x) = f  5 For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described. 63. The graph of f (x) = ∣ x ∣ is reflected over the y-axis ...
8 10 x Figure 32 78. g(x) = f (x) − 2 79. g(x) = −f (x) 80. g(x) = f (x + 1) 81. g(x) = f (x − 2) SECTION 1.6 aBsolute value Functions 89 leARnInG OBjeCTIVeS In this section you will: • • • Graph an absolute value function. Solve an absolute value equation. Solve an absolute value inequality. 1. 6 ABSOlUTe VAlUe FUnCT...
� x − 5 ∣ ≤ 4 is equivalent to 1 ≤ x ≤ 9. However, mathematicians generally prefer absolute value notation. 90 CHAPTER 1 Functions Try It #1 Describe all values x within a distance of 3 from the number 2. Example 2 Resistance of a Resistor Electrical parts, such as resistors and capacitors, come with specified values o...
5 Solution The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure 6. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 –4 –3 –2 21 3 4 5 x (3, –2) Figure 6 We also notice that the graph appears vertically stretc...
or two points (see Figure 8). y 5 4 3 2 1 –1 –1 –2 –3 –4 (a) –5 –4 –3 –2 21 3 4 5 x –5 –4 –3 –2 y 5 4 3 2 1 –1–1 –2 –3 –4 (b) 21 3 4 5 x –5 –4 –3 –2 y 5 4 3 2 1 –1–1 –2 –3 –4 (c) 21 3 4 5 x Figure 8 (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the h...
. Solution 0 = \| 4x + 1 \| − 7 7 = \| 4x + 1 \| 7 = 4x + 1 or −7 = 4x + 1 6 = 4x 6 __ x = = 1.5 4 −8 = 4x −8 ___ 4 x = = −2 Substitute 0 for f (x). Isolate the absolute value on one side of the equation. Break into two separate equations and solve. The function outputs 0 when x = 1.5 or x = −2. See Figure 9. y 10 8 6 4 2 ...
2\| + 2 f (x) = 1 x 21 3 4 5 –5 –4 –3 –2 10 8 6 4 2 –1–2 –4 –6 –8 –10 Figure 10 Try It #5 Find where the graph of the function f (x) = −\| x + 2 \| + 3 intersects the horizontal and vertical axes. Solving an Absolute Value Inequality Absolute value equations may not always involve equalities. Instead, we may need to solv...
2. Test intervals created by the boundary points to determine where \| x − A \| ≤ B. 3. Write the interval or union of intervals satisfying the inequality in interval, inequality, or set-builder notation. Example 6 Solving an Absolute Value Inequality Solve \| x − 5\| < 4. Solution With both approaches, we will need to kn...
are less than the output values of g(x). • The absolute value is less than or equal to 4 between these two points, when 1 ≤ x ≤ 9. In interval notation, this would be the interval [1, 9]. Analysis For absolute value inequalities, \| x − A \| < C, \| x − A \| > C, −C < x − A < C, x − A < −C or x − A > C. The < or > symbol ...
25 21 –5 –2 –4 Below x-axis –3 –1–1 –2 –3 –4 –5 Figure 12 x = −2.75 x 3 4 5 Below x-axis 1 11 __ __ We observe that the graph of the function is below the x-axis left of x = − and right of x = 4 4 function values are negative to the left of the first horizontal intercept at x = − 1 __, and negative to the right of the ...
x is from 10 is at least 15 units. Express this using absolute value notation. 9. Find all function values f (x) such that the distance from f (x) to the value 8 is less than 0.03 units. Express this using absolute value notation. For the following exercises, solve the equations below and express the answer using set ...
the given functions by hand. 40. y = \| x \| − 2 41. y = −\| x \| 42. y = −\| x \| − 2 43. y = −\| x − 3 \| − 2 44. f (x) = −\| x − 1 \| − 2 45. f (x) = −\| x + 3 \| + 4 46. f (x) = 2\| x + 3 \| + 1 47. f (x) = 3\| x − 2 \| + 3 48. f (x) = \| 2x − 4 \| − 3 49. f (x) = \| 3x + 9 \| + 2 50. f (x) = −\| x − 1 \| − 3 51. f (x) = −\| x + 4 \| −3 ...
using absolute value notation. 61. The true proportion p of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation. 62. Students who score within 18 points of the number 82 will pass a particular test. Write this statement using abs...
) 9 5 __ (75 − 32) ≈ 24°C. 9 Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, he sends his assistant the week’s weather forecast from Figure 2 for Milan, and asks her to convert all of the temperatures to degrees Fahrenheit. Mon Tue Web Thu 26°C \| 19°C 29°C \| 19°C 30°C \| 20°C 26°C \| 18°C Fi...
have a reciprocal, some functions do not have inverses. Given a function f (x), we can verify whether some other function g (x) is the inverse of f (x) by checking whether either g ( f (x)) = x or f (g (x)) = x is true. We can test whichever equation is more convenient to work with because they are logically equivalen...
, then f −1 (4) = 2; f (5) = 12, then f −1 (12) = 5. Alternatively, if we want to name the inverse function g, then g (4) = 2 and g (12) = 5. Analysis Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed. See Table 1. (x, f (x)) (2, 4) (5, 12) (x, g (x)) (4, 2) (12, 5) ...
exponent, Try It #3 If f (x) = (x − 1)3 and g (x) = 3 √ — x + 1, is g = f −1? Finding Domain and Range of Inverse Functions The outputs of the function f are the inputs to f −1, so the range of f is also the domain of f −1. Likewise, because the inputs to f are the outputs of f −1, the domain of f is the range of f −1...
with its domain limited to [0, ∞), which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function). If f (x) = (x − 1)2 on [1, ∞), then the inverse function is f −1(x) = √ x + 1. — • The domain of f = range of f −1 = [1, ∞). • The domain of f −1 = range of f = [0...
1 _ x2 f (x) = 3 √ — x f (x) = √ — x f (x) = ∣x∣ Table 2 Solution The constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no meaningful inverse. The absolute value function can be restricted to the domain [0, ∞), where it is ...
(minutes) f (t) (miles) 30 20 50 40 70 60 90 70 Table 3 Solution The inverse function takes an output of f and returns an input for f. So in the expression f −1(70), 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function f, 90 min...
5 Solution To evaluate g(3), we find 3 on the x-axis and find the corresponding output value on the y-axis. The point (3, 1) tells us that g(3) = 1. To evaluate g −1(3), recall that by definition g −1(3) means the value of x for which g(x) = 3. By looking for the output value 3 on the vertical axis, we find the point ...
4. Solution y = + 4 Set up an equation. 2 ____ x − 3 2 _____ x − 3 2 ____ y − 4 2 ____ = Subtract 4 from both sides. Multiply both sides by x − 3 and divide by y − 4. x = + 3 Add 3 to both sides. So f −1 (y) = + 3 or f −1 (x) = 2 _____ y − 4 2 ____ x − 4 + 3. Analysis The domain and range of f exclude the values 3 and...
atic function with domain restricted to [0, ∞). Restricting the domain to [0, ∞) makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain. We already know that the inverse of the toolkit quadratic function is the square root function, that is, f −1(...
function and its inverse, showing reflection about the identity line Try It #9 Draw graphs of the functions f and f −1 from Example 8. Q & A… Is there any function that is equal to its own inverse? Yes. If f = f −1, then f (f (x)) = x, and we can think of several functions that have this property. The identity functio...
(x) = x2 − 5 16. Given f (x) = and g(x) = x __ 2 + x 2x _____ : 1 − x a. Find f (g(x)) and g (f (x)). b. What does the answer tell us about the relationship between f (x) and g(x)? For the following exercises, use function composition to verify that f (x) and g(x) are inverse functions. 17. f (x) = 3 √ — x − 1 and g (...
Figure 12 of f. nUMeRIC For the following exercises, evaluate or solve, assuming that the function f is one-to-one. 33. If f (6) = 7, find f −1(7). 34. If f (3) = 2, find f −1(2). 35. If f −1(−4) = −8, find f (−8). 36. If f −1(−2) = −1, find f (−1). For the following exercises, use the values listed in Table 6 to eval...
interval absolute minimum the lowest value of a function over an interval absolute value equation an equation of the form ∣A∣ = B, with B ≥ 0; it will have solutions when A = B or A = −B absolute value inequality a relationship in the form ∣A∣ < B, ∣A∣ ≤ B, ∣A∣ > B, or ∣A∣ ≥ B average rate of change the difference in ...
the domain of f; this also implies that f ( f −1(x)) = x for all x in the domain of f −1 local extrema collectively, all of a function’s local maxima and minima local maximum a value of the input where a function changes from increasing to decreasing as the input value increases. local minimum a value of the input whe...
(x1) _ x2 − x1 Composite function (f ∘ g)(x) = f (g(x)) Vertical shift g(x) = f (x) + k (up for k > 0) Horizontal shift g(x) = f (x − h) (right for h > 0) Vertical reflection g(x) = −f (x) Horizontal reflection g(x) = f (−x) Vertical stretch g(x) = af (x) (a > 0) Vertical compression g(x) = af (x) (0 < a < 1) Horizont...
an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number. • The domain of a function can be determined by listing the input values of a set of ordered pairs. See Example 1. • The domain of a function can also be determined by identifying the input values of a functio...
ima and minima. See Example 10. 1.4 Composition of Functions • We can perform algebraic operations on functions. See Example 1. • When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function. • The function produced by combining two functions is a composite func...
the origin. • Even functions satisfy the condition f (x) = f (−x). • Odd functions satisfy the condition f (x) = −f (−x). • A function can be odd, even, or neither. See Example 12. • A function can be compressed or stretched vertically by multiplying the output by a constant. See Example 13, Example 14, and Example 15...
its domain. • For a tabular function, exchange the input and output rows to obtain the inverse. See Example 5. • The inverse of a function can be determined at specific points on its graph. See Example 6. • To find the inverse of a formula, solve the equation y = f (x) for x as a function of y. Then exchange the label...
graph the functions. 12. f (x) = ∣x + 1∣ 13. f (x) = x 2 − 2 CHAPTER 1 review 119 For the following exercises, use Figure 2 to approximate the values. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 –4 –3 –2 14. f (2) 15. f (−2) 21 3 4 5 x 16. If f (x) = −2, then solve for x. 17. If f (x) = 1, then solve for x. Figure 2 For the foll...
function graphed in Exercise 28. 120 CHAPTER 1 Functions 32. For the graph in Figure 3, the domain of the function is [−3, 3]. The range is [−10, 10]. Find the absolute minimum of the function on this interval. 33. Find the absolute maximum of the function graphed in Figure 3. –5 –4 –3 –2 y 10 8 6 4 2 –1 –2 –4 –6 –8 –...
+ 5 50. f (x) = 5 √ — −x − 4 For the following exercises, sketch the graph of the function g if the graph of the function f is shown in Figure 4. y 5 4 3 2 1 –1 –1 –2 –5 –4 –3 –2 21 3 4 5 x Figure 4 53. g(x) = f (x − 1) 54. g(x) = 3f (x) CHAPTER 1 review 121 For the following exercises, write the equation for the stan...
= \| x − 5 \| 67. f (x) = −\| x − 3 \| 68. f (x) = \| 2x − 4 \| 122 CHAPTER 1 Functions For the following exercises, solve the absolute value equation. 69. \| x + 4 \| = 18 x + 5  =  3 70.  1 x − 2  __ __ 4 3 For the following exercises, solve the inequality and express the solution using interval notation. 71. \| 3x − 72....
notation. 7. Given f (x) = 2x 2 − 5x, find f (a + 1) − f (1). 8. Graph the function f (x) = { x + 1 if −2 < x < 3 x ≥ 3 −x if 9. Find the average rate of change of the function f (x) = 3 − 2x 2 + x by finding f (b) − f (a) _ b − a. For the following exercises, use the functions f (x) = 3 − 2x 2 + x and g(x) = √ — x to...
an ordered pair. Figure 1 For the following exercises, use the graph of the piecewise function shown in Figure 2. 27. Find f (2). 28. Find f (−2). 29. Write an equation for the piecewise function. y 5 4 3 2 1 f 21 3 4 5 x –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 Figure 2 For the following exercises, use the values listed in Tabl...
or constant. • • • Calculate and interpret slope. Write the point-slope form of an equation. Write and interpret a linear function. 2.1 lIneAR FUnCTIOnS Figure 1 Shanghai Maglev Train (credit: “kanegen”/Flickr) Just as with the growth of a bamboo plant, there are many situations that involve constant change over time....
/maglev-train.htm SECTION 2.1 linear Functions 127 Representing a Linear Function in Function Notation Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where x is the input va...
200 100 0 1 2 3 4 Time (s) 5 Figure 3 The graph of D(t) = 83t + 250. Graphs of linear functions are lines because the rate of change is constant. Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line f (x) = 2x + 1. Ask yourself what numbers can be in...
positive slope slants upward from left to right as in Figure 5(a). For a decreasing function, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in Figure 5(b). If the function is constant, the output values are the same fo...
. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after x days. c. The cost function can be represented as f (x) = 50 because the number of days does not affect the total cost. The slope is 0 so the function is constan...
A… Are the units for slope always units for the output __? units for the input Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input...
27,800 − 23,400 = 4,400 people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years. 4,400 people __ = 4 years 1,100 people __ year So the population increased by 1,100 people per year. Analysis Because we are told that the population increased,...
linear Functions Writing the Equation of a Line Using a Point and the Slope The point-slope form is particularly useful if we know one point and the slope of a line. Suppose, for example, we are told hat a line has a slope of 2 and passes through the point (4, 1). We know that m = 2 and that x1 = 4 and y1 = 1. We can ...

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.