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5000 feet north of the intersection. The runner runs due west at 17 feet per second. When will the runner’s feet get wet? 31 rider 100 feet ground level 62 ft. tower 60 feet operator 24 feet (a) Impose a coordinate system. (b) Suppose a rider is located at the point in the picture, 100 feet above the ground. If the ri... |
Ballard is 8 miles South and 1 mile West of Edmonds. Impose coordinates with Ballard as the origin. sailboat Kingston Edmonds North Ballard UDub (a) Find the equations for the lines along which the ferry is moving and draw in these lines. (b) The sailboat has a radar scope that will detect any object within 3 miles of... |
Linear Modeling Sometimes, we will begin a section by looking at a specific problem which highlights the topic to be studied; this section offers the first such vista. View these problems as illustrations of precalculus in action, rather than confusing examples. Don’t panic, the essential algebraic skills will be review... |
dollars) S = (1987,28313) R = (1970,5616) x-axis (yeara) Data points for women. (b) Data points for men. Figure 4.1: Visualizing the data. then the point (x, y) lies on the line connecting P and Q. Likewise, for men, if the earning power in year x is $y, then the point (x, y) lies on the line connecting R and S. In the... |
graphs would begin with the simple cases, gradually working toward the more complicated. Thinking visually, the simplest curves in the plane would be straight lines. As we discussed in Chapter 3, a point on the vertical line in Figure 4.3(a) will always have the same x-coordinate; we refer to this line as the graph of... |
to Q = x2 − x1. ∆y = change in y going from P to Q = y2 − y1. We define the slope of the line ℓ to be the ratio of ∆y by ∆x, which is usually denoted by m: m def= = slope of ℓ = ∆y ∆x y2 − y1 x2 − x1 change in y change in x (4.3) Notice, we are using the fact that the line is non-vertical to know that this ratio is alw... |
−∆x. 37 Q = (4,5) 5 1 P = (1,1) ∆y = 4 ∆x = 3 1 4 Figure 4.4: Computing the slope of a line. We CANNOT reverse the order in just one of the calculations and get the same slope: m = y2 − y1 x2 − x1 6 = y2 − y1 x1 − x2, and m = y2 − y1 x2 − x1 6 = y1 − y2 x2 − x1.!!! CAUTION!!! It is very important to notice that the ca... |
4) (4.5) Equation 4.4 is usually called the point slope formula for the line ℓ (since the data required to write the equation amounts to a point (x1,y1) on the line and the slope m), whereas Equation 4.5 is called the two point formula for the line ℓ (since the data required amounts to the coordinates of the points P a... |
− 1) + 1 |x is any real number Returning to the general situation, we can obtain a third general equation for a non-vertical line. To emphasize what is going on here, plug the specific value x = 0 into Equation 4.4 and obtain the point R = (0,b) on the line, where b = m(0 − x1) + y1 = −mx1 + y1. But, Equation (4.4) can... |
will be of the most interest to us, tions. since these are the lines that can be viewed as the graphs of functions; we will discuss this in Chapter 5. 4.5 Lines and Rate of Change If we draw a non-vertical line in the xy coordinate system, then its slope will be the rate of change of y with respect to x: slope = = ∆y ... |
moving object. Suppose an object moves along a straight line at a constant speed m: See Figure 4.7. If we specify a reference point, we can let b be the starting location of the moving object, which is usually called the initial location of the object. We can write down an equation relating the initial location b, the... |
yells “go!” and both Mookie and Asia start running directly away from Linda to catch a tossed frisbee. Find linear equations for the distances between Linda, Mookie and Asia after t seconds. Linda Mookie Asia Solution. Let sM be the distance between Linda and Mookie and sA the distance between Linda and Asia, after t ... |
21. (4.6) The graph of this line will pass through the two points R and S in Figure 4.2. We sketch the graph in Figure 4.9, indicating two new points T and U. We can use the model in (4.6) to make predictions of two different sorts: (i) predict earnings at some date, or (ii) predict when a desired value for earnings wi... |
on men? You will also be asked to think about this question in the exercises. 4.7 What’s Needed to Build a Linear Model? As we progress through this text, a number of different “types” of mathematical models will be discussed. We will want to think about the information needed to construct that particular kind of math... |
Facts 4.9.1. Two non-vertical lines in the plane are parallel precisely when they both have the same slope. Two non-vertical lines are perpendicular precisely when their slopes are negative reciprocals of one another. 4.10. INTERSECTING CURVES II 45 Example 4.9.2. Let ℓ be a line in the plane passing through the point... |
technique for finding the mula. shortest distance between a “line” and a “point.” 46 CHAPTER 4. LINEAR MODELING North flight path irrigated field West Q East s e l i m 2 crop duster P 1.5 miles South Figure 4.11: The flight path of a crop duster. Example 4.10.2. A crop dusting airplane flying a constant speed of 120 mph is... |
nd x = −1, 0.2195. Conclude that the x coordinate of P is 0.2195. To find the y coordinate, plug into the line equation and get y = −0.9756. Conclude that P = (0.2195, −0.9756) 4.11. UNIFORM LINEAR MOTION 47 3. Find the distance from P to Q by using the distance formula: d = (−1 − 0.2195)2 + (0 − (−0.9756))2 = 1.562 mil... |
the xy-plane. He runs in a straight line from the point (2,3) to the point (5, − 4), taking 6 seconds to do so. Find his equations of motion. 48 CHAPTER 4. LINEAR MODELING Solution. We begin by setting a reference for our time parameter. Let’s let t = 0 represent the instant when Bob is at the point (2,3). In this way... |
, y = 3 − 7 6 (30) = −32. Example 4.11.2. Olga is running in the xy-plane, and the coordinate are given in meters (so, for example, the point (1,0) is one meter from the origin (0,0)). She runs in a straight line, starting at the point (3,5) and running along the line y = − 1 3 x + 6 at a speed of 7 meters per second, ... |
ga’s location t seconds after she starts is (x,y) where x = a + bt and y = c + dt. When t = 0, x = 3, and y = 5, so x = 3 = a + b(0) = a and y = 5 = c + d(0) = c and so a = 3 and c = 5. Also, when t = 0.45175395, x = 6 and y = 4, so x = 6 = a+b(0.45175395) = 3+b(0.45175395) and y = 4 = c+d(0.45175395) = 5+d(0.45175395)... |
) Passing through the point (−1, − 2) with slope m = 40. (c) With y-intercept b = −2 and slope m = −2. (d) Passing through the point (4,11) and having slope m = 0. (e) Perpendicular to the line in (a) and pass- ing through (1,1). (f) Parallel to the line in (b) and having y- intercept b = −14. (g) Having the equation 3... |
in Seattle and Port Townsend in 1983 and 1998? (e) When will the average sales price in Seattle and Port Townsend be equal and what is this price? (f) When will the average sales price in Port Townsend be $15,000 less than the Seattle sales price? What are the two sales prices at this time? (g) When will the Port Town... |
b) Where is Allyson when the bungee reaches its maximum length? 4.13. EXERCISES 53 20 ft Building 30 ft Allyson Adrian start Problem 4.8. Dave is going to leave academia and go into business building grain silos. A grain silo is a cylinder with a hemispherical top, used to store grain for farm animals. Here is a 3D vie... |
of Paris. Florence is 25 miles East, and 45 miles North of Rome. On her trip, how close does Pam get to Paris? Problem 4.11. Angela, Mary and Tiff are all standing near the intersection of University and 42nd streets. Mary and Tiff do not move, but Angela runs toward Tiff at 12 ft/sec along a straight line, as picture... |
, but 2 seconds after she does. How long does it take Mercutio to reach the y-axis? Problem 4.15. (a) Solve for x. hose N S E W 100 ft (b) Solve for t: 2 = (1 + t)2 + (1 − 2t)2. (c) Solve for t: 3 √5 p = (1 + t)2 + (1 − 2t)2. (d) Solve for t: 0 = p (1 + t)2 + (1 − 2t)2. p 20 ft Problem 4.16. (a) Solve for x: 100 ft. si... |
these three descriptive modes, imagine we have tabulated (Figure 5.2) the height of the gull above cliff level at one-second time intervals 55 56 CHAPTER 5. FUNCTIONS AND GRAPHS for a 10 second time period. Here, a “negative height” means the gull is below cliff level. We can try to visualize the meaning of this data ... |
, look at this equation involving the variables t and s: s = 15 8 (t − 4)2 − 10. 5.2. WHAT IS A FUNCTION? 57 If we plug in t = 0, 1, 2, 9, 10, then we get s = 20, 6.88, −2.5, 36.88, 57.5, respectively; this was some of our initial tabulated data. This same equation produces ALL of the data points for the other two plot... |
produce such a “procedure” using data, plots of curves and equations. y-axis P = (x,y) y x x-axis • A table of data, by its very nature, will relate two columns of data: The output and input values are listed as column entries of the table and reading across each row is the “procedure” which relates an input with a un... |
a unique) y value. We call the mathematical expression f(x) ”the rule”. • A set D of x-values we are allowed to plug into f(x), called the ”domain” of the function. 5.2. WHAT IS A FUNCTION? 59 • The set R of output values f(x), where x varies over the domain, called the ”range” of the function. Any time we have a funct... |
horizontal axis. (iv) Consider the equation y = 1 x, then the rule f(x) = 1 x defines a function, as long as we do not plug in x = 0. For example, take the domain to be the non-zero real numbers. (v) Consider the equation y = √1 − x2. Before we start plugging in x values, we want to know the expression under the radica... |
to try and solve this equation for y in terms of x, you’d first write y2 = 1 − x and then take a square root (to isolate y); but the square root introduces TWO roots, which is just another way of reflecting the fact there can be two y values attached to a single x value. Alternatively, you can solve the equation for x i... |
center of a basketball as you dribble, depending on time. Let s be the height of the basketball center at time t seconds after you start dribbling. Given a time t, if we freeze the action, the center of the ball has a single unique height above the floor, call it h(t). So, the height of the basketball center is given b... |
... x y 5 3 1 -1... −2x + 3 point (x,y) (-1,5) (0,3) (1,1) (2,-1)... (x, − 2x + 3) y-axis Graph of y = −2x + 3. x-axis (a) Tabulated data. (b) Visual data. Figure 5.7: Symbolic versus visual view of data. · This tells us that the points (0, 3), (1, 1), (2, −1), (−1, 5) are solutions of the equation y = −2x + 3. For exa... |
, namely f(x). This means the point P = (x, f(x)) is the ONLY solution to the equation y = f(x) with first coordinate x. We define the graph of the function y = f(x) to be the plot of all solutions of this equation (in the xy coordinate system). It is common to refer to this as either the “graph of f(x)” or the “graph of... |
way to test if a curve is the graph of a function. Important Procedure 5.4.1. The vertical line test. Draw a curve in the xy-plane and specify a set D of x-values. Suppose every vertical line through a value in D intersects the curve exactly once. Then the curve is 6 64 CHAPTER 5. FUNCTIONS AND GRAPHS the graph of som... |
would 1 write this constraint as 0 2. t ≤ ≤ 5.5 Linear Functions A major goal of this course is to discuss several different kinds of functions. The work we did in Chapter 4 actually sets us up to describe one 5.6. PROFIT ANALYSIS 65 very useful type of function called a linear function. Back in Chapter 4, we discusse... |
11: Distance functions. For the rule s(t), the best domain would again be 0 2. We have graphed these two functions in the same coordinate system: See Figure 5.11 (Which function goes with which graph?). ≤ ≤ t 5.6 Profit Analysis Let’s give a first example of how to interpret the graph of a function in the context of an a... |
100 e s n e p x e 1500 1000 500 (x, e(x)) Q x-axis (units sold) 20 40 60 80 100 (a) Gross income graph. (b) Expenses graph. Figure 5.12: Visualizing income and expenses. things: x on the horizontal axis; a point on the graph of the gross income or expense function; • • y on the vertical axis. • If x = 20 units sold, t... |
+ 10000 = 0. Applying the quadratic formula, we get two answers: x = 0.78 or 57. Now, we face a problem: Which of these two solutions is the answer to the original problem? We are going to argue that only the second solution x = 57 gives us the break even point. What about the other ”solution” at x = 0.78? Try pluggin... |
is v(20)? What are the smallest and largest values of v(x) on the domain 0 20? ≤ x ≤ ≤ y-axis 60 40 20 (0,4) g(x) (20,60) (0,24) f(x) (20,20) 10 20 x-axis Problem 5.3. Dave leaves his office in Padelford Hall on his way to teach in Gould Hall. Below are several different scenarios. In each case, sketch a plausible (rea... |
that keeps track of Dave’s distance s from Gould Hall at time t. How would your graphs change in (a)-(e)? Problem 5.4. At 5 AM one day, a monk began a trek from his monastery by the sea to the monastery at the top of a mountain. He reached the mountain-top monastery at 11 AM, spent the rest of the day in meditation, 5... |
,7 − 3√3). (e) Find the exact coordinates of the point (x,y) on the graph with x = 1 + √2. p Problem 5.7. After winning the lottery, you decide to buy your own island. The island is located 1 km offshore from a straight portion of the mainland. There is currently no source of electricity on the island, so you want to r... |
fix” the graph, you are allowed to cut the curve into pieces and such that each piece is the graph of a function. Many of these problems have more than one correct answer. Problem 5.10. Find an EXACT answer for each problem. (a) Solve for = 30 x2 − 4x − 21 (b) Solve for x √5x − 4 = x 2 + 2 (c) Solve for x √x + √x − 20 =... |
Visualizing the domain and range A function is a package that consists of a rule y = f(x), a domain of allowed x-values and a range of output y-values. The domain can be visualized as a subset of the x-axis and the range as a subset of the y-axis. If you are handed the domain, it is graphically easy to describe the ra... |
. Keeping track of this information on a number line is called a sign plot for the function. We include a “shadow” of the graph in Figure 6.3 6.1. VISUAL ANALYSIS OF A GRAPH 75 positive positive negative negative x-axis Figure 6.3: Sign plot. to emphasize how we arrived at our “positive” and “negative” labeling of the ... |
-intercept: See Figure 6.5 The graph of a function y = f(x) crosses the vertical line x = h at the point (h,f(h)). To find where the graph of a function y = f(x) crosses the horizontal line y = k, first solve the equation k = f(x) for x. If the equation k = f(x) has solutions x1, x2, x3, x4, then the points of intersecti... |
or vice versa; these “peaks” and “valleys” are called local maxima and local minima. Some folks refer to either case as a local extrema. People have invested a lot of time (centuries!) and energy (lifetimes!) into the study of how to find local extrema for particular function graphs. We will see some basic examples in ... |
could rewrite this as (y − k)2 = r2 − (x − h)2, then take the square root of each side. However, the resulting equivalent equation would be y = k r2 − (x − h)2 ± and the presence of that ± r2 − (x − h)2 or r2 − (x − h)2. sign is tricky; it means we have two equations: Each of these two equations defines a function: p f... |
cross section of the tunnel; by symmetry the walkway is centered about the origin. With this coordinate system, the graph of the equation x2 + y2 = 152 = 225 will be the circular cross-section of the tunnel. In the case of Deck A, we basically need to determine how close to each edge of the tunnel a 6 foot high person... |
to the right and left of the centerline. 6.3 Multipart Functions So far, in all of our examples we have been able to write f(x) as a nice compact expression in the variable x. Sometimes we have to work harder. 80 CHAPTER 6. GRAPHICAL ANALYSIS As an example of what we have in mind, consider the graph in Figure 6.11(a):... |
we would find which of the five cases covers x = 3.56, then apply that part of the rule to compute f(3.56) = 1. Our first multipart function example illustrated how to go from a graph in the plane to a rule for f(x); we can reverse this process and go from the rule to the graph. Example 6.3.1. Sketch the graph of the mul... |
. Three portions of the graph are decreasing and two portions are increasing. Why doesn’t the graph touch the t axis? ≤ ≤ t Figure 6.13: Dribbling. 82 CHAPTER 6. GRAPHICAL ANALYSIS 6.4 Exercises Problem 6.1. The absolute value function is defined by the multipart rule: Problem 6.4. (a) Let f(x) = x + |2x − 1|. Find all ... |
< 3, 3. 4 − x if x ≥ h(x) = −8 − 4x if x 1 + 1 −2, 3 x if x > −2. ≤ Problem 6.6. Pizzeria Buonapetito makes a triangular-shaped pizza with base width of 30 inches and height 20 inches as shown. Alice wants only a portion of the pizza and does so by making a vertical cut through the pizza and taking the shaded portion.... |
and determine the range. (b) Steve drives from Spokane to Bellevue at 70 mph, departing from Spokane at 12:00 noon. Find a function s(t) for his distance from Bellevue at time t. Sketch the graph, specify the domain and determine the range. (c) Find a function d(t) that computes the distance between Joan and Steve at ... |
Simply as far as possible R R vertical cross-section Here, R indicates a circle of radius 10 feet and all of the indicated circle centers lie along the common horizontal line 10 feet above and parallel to the ditch bottom. Assume that water is flowing into the ditch so that the level above the bottom is rising 2 inches... |
example, then show every other standard parabola can be obtained from it via some specific geometric maneuvers. • As we perform these geometric maneuvers, we keep track of how the function equation for the curve is changing. This discussion will amount to a concrete application of a more general set of tools developed ... |
it not at all obvious why the vertex form is obtained by this equality: −3x2 + 6x − 1 = −3(x − 1)2 + 2. The reason behind this equality is the technique of completing the square. In the end, we will almost always be interested in the vertex form of a quadratic. This is because a great deal of qualitative information a... |
= x2. Using a graphing device, we can check that the corresponding equations for the dashed graphs would be ≤ ≤ x y = (x − (−2))2 = (x + 2)2 = x2 + 4x + 4, which is the plot with lowest point (−2, 0) and y = (x − (−4))2 = (x + 4)2 = x2 + 8x + 16, which is the plot with lowest point (−4, 0). In general, if h is negativ... |
2 Second Maneuver: Reflection Next, we can reflect any of the curves y = p(x) obtained by horizontal or vertical shifting across the x-axis. This procedure will produce a new curve which is the graph of the new function y = −p(x). For example, begin with the four dashed curves in the previous figure. Here are the reflected... |
+ k, for some constants a, h, and k and a = 0. The vertex of the parabola is (h, k) and the axis of symmetry is the line x = h. If a > 0, then the parabola opens upward; if a < 0, then the parabola opens downward. 6 6 90 CHAPTER 7. QUADRATIC MODELING Example 7.1.2. Describe a sequence of geometric operations leading f... |
the skinny parabola opening upward with vertex (1, 0). A reflection yields the graph of y = −3(x − 1)2; this is the downward opening parabola with vertex (1, 0). A vertical shift by k = 2 yields the graph of y = −3(x − 1)2 + 2; this is the downward opening parabola with vertex (1, 2). 7.2 Completing the Square By now i... |
coefficients. can proceed to solve for these: −3 = a 6 = −2ah −1 = ah2 + k The first equation just hands us the value of a = −3. Next, we can plug this value of a into the second equation, giving us 6 = −2ah = −2(−3)h = 6h, 92 CHAPTER 7. QUADRATIC MODELING so h = 1. Finally, plug the now known values of a and h into the... |
shifting by k = 3.562 gives y = f(x) = −4(x − 0.625)2 + 3.562. • • • • Example 7.2.3. A drainage canal has a cross-section in the shape of a parabola. Suppose that the canal is 10 feet deep and 20 feet wide at the top. If the water depth in the ditch is 5 feet, how wide is the surface of the water in the ditch? 7.3. I... |
achieved is the first coordinate of the vertex. As we know, it is easy to read off the vertex coordinates when a quadratic function is written in vertex form. If instead we are given a quadratic function y = ax2 + bx + c, we can use the technique of completing the square and arrive at a formula for the coordinates of t... |
coordinate of the ball at time t. The function y(t) is a quadratic function. If we had this function in hand, we could determine when the ball hits the ground by solving the equation 0 = y(t), but we would not be able to determine where the ball hits the ground. A second approach is to forget about the time variable an... |
(48)2 − 4 (−16) (50) = 3.818 sec or − 0.818 sec p 2 16 · Conclude the ball hits the ground after 3.818 seconds. Finally, the height of the cliff is the height of the ball zero seconds after release; i.e., y(0) = 50 feet is the height of the cliff. Here are two items to consider carefully: 1. The graph of y(t) is NOT t... |
500 1000 height of balloon above lake level A B height of balloon above ground level height of ground above lake level Figure 7.16: The height of the balloon y as a function of x. The function y = f(x) keeps track of the height of the balloon above lake level at a given x location on the horizontal axis. The line ℓ wi... |
875,87.5). Next, we want to study the height of the balloon above the ground. Let y = g(x) be the function which represents the height of the balloon above 98 CHAPTER 7. QUADRATIC MODELING the ground when the horizontal coordinate is x. We find g(x) = height of the balloon above lake level with horizontal coordinate x −... |
inear points in the plane such that the x-coordinates are all different. Then there exists a unique standard parabola passing through these three points. This parabola is the graph of a quadratic function y = f(x) = ax2 + bx + c and we can find these coefficients by simultaneously 7.4. QUADRATIC MODELING PROBLEMS 99 vert... |
will follow a typical practice in real estate and use the units of K, where K = $1,000. For example, a house valued 100 CHAPTER 7. QUADRATIC MODELING at $235,600 would be worth 235.6 K. These will be the units we use, which essentially saves us from drowning in a sea of zeros! We are given three pieces of information ... |
26) = 9 20(26)2 − 3 226 + 50 = 315.2; i.e., the value of the house is $315,200. To find when the house will be worth $1,000,000, we note that $1,000,000 = 1,000 K and need to solve the equation 1000 = v(x) = 0 = 9 20 9 20 x2 − x2 − 3 2 3 2 x + 50 x − 950. 7.5. WHAT’S NEEDED TO BUILD A QUADRATIC MODEL? 101 By the quadrat... |
) point on the graph. This highest (or lowest) point is known as the vertex of the graph; its location is given by (h,k) where h = − b 2a and k = f(h). If a > 0, then the vertex is the lowest (or minimum) point on the graph, and the parabola ”opens upward”. If a < 0, then the vertex is the highest (or maximum) point on... |
function g(x) = −(x + 3)2 + 3 on the interval 0 4. What is the maximum value of g(x) on that interval? What is the minimum value of g(x) on that interval? ≤ ≤ x Problem 7.4. If the graph of the quadratic function f(x) = x2 + dx + 3d has its vertex on the x-axis, what are the possible values of d? What if f(x) = x2 + 3... |
point (1,2) lie on the graph of y = f(x)? Why or why not? (b) If b is a constant, where does the line y = 1 + 2b intersect the graph of y = x2 + bx + b? 104 CHAPTER 7. QUADRATIC MODELING (c) If a is a constant, where does the line y = 1 − a2 intersect the graph of y = x2 − 2ax + 1? (d) Where does the graph of y = −2x2... |
wire 60 inches long and cuts it into two pieces. Steve takes the first piece of wire and bends it into the shape of a perfect circle. He then proceeds to bend the second piece of wire into the shape of a perfect square. Where should Steve cut the wire so that the total area of the circle and square combined is as small... |
equations, find the value(s) of the constant α so that the equation has exactly one solution, and determine the solution for each value. (−50,275) beet juice (200,300) soy milk (400,50) Tina Michael Problem 7.18. Consider the equation: αx2 + 2α2x + 1 = 0. Find the values of x that make this equation true (your answer w... |
) Flash at t = 1. oxygen rate 1 hr 2 4 6 hours 8 10 (b) Flash at t = 5. Figure 8.1: Light flashes. Suppose we want to model the oxygen consumption when a green light pulse occurs at time t = 5 (instead of time t = 1), what is the mathematical model? For starters, it is pretty easy to believe that the graph for this ne... |
working with this kind of setup. Abstractly, we have just described a situation where we take two functions and build a new 8.1. THE FORMULA FOR A COMPOSITION 109 function which “composes” the original ones together; schematically the situation looks like this: Example 8.1.1. A pebble is tossed into a pond. The radius... |
x) and that additionally the independent variable x is itself a function of a different independent variable t; i.e., x = g(t). Then we can replace every occurrence of “x” in f(x) by the expression “g(t),” thereby obtaining y as a function in the independent variable t. We usually denote this new function of t: y = f(g... |
+ 2t ♥ + 2. ♥ It is natural to ask: What good is this whole business about compositions? One way to think of it is that we can use composite functions to break complicated functions into simpler parts. For example, y = h(x) = x2 + 1 p can be written as the composition f(g(x)), where y = f(z) = √z and z = g(x) = x2+1. ... |
us: f(g(x)) = f(2x + 1) = (2x + 1)2. Here are three other examples: If f(x) = √x, g(x) = 2x2 + 1, then f(g(x)) = √2x2 + 1. If f(x) = 1 x, g(x) = 2x + 1, then f(g(x)) = 1 2x+1. If f(x) = x2, g(x) = △ − x, then f(g(x)) = 2 − 2x △ △ + x2. • • • 112 y-axis 10 8 6 4 2 x-axis −3 −2 −1 1 2 3 Figure 8.4: Sketching composite f... |
domain of allowed input values, and a range of output values. When we start to compose functions, we sometimes need to worry about how the domains and ranges of the composing functions affect the composed function. First off, when you form the composition f(g(x)) of f(x) and g(x), the range values for g(x) must lie wi... |
f(g(x)) = x2 −2x+1 on the domain 0 x ≤ ≤ 2. Example 8.2.3. Let y = f(z) = √z, z = g(x) = x + 1. What is the largest possible domain so that the composition f(g(x)) makes sense? 114 CHAPTER 8. COMPOSITION y range y = f(z) z domain z z = g(x) = x + 1 −1 desired range x required domain (a) y = √z. (b) z = x = 1. Figure 8... |
t + 3 if 1 if 3 1 t t ≤ ≤ ≤ 3 ≤ Now, suppose instead we apply the flash of high intensity green light at the time t = 5. Verify that the mathematical model for this experiment is given by f(g(t)), where g(t) = t − 4. 8.2. DOMAIN, RANGE, ETC. FOR A COMPOSITION 115 Solution. Our expectation is that the plot for this new... |
t2 − 8t + = when 5 t ≤ ≤ 7. t. We now appeal to Procedure 8.2.1 and just replace every occurrence of t in this function by g(t). That gives us this NEW domain condition and function equation: ≤ 2 3 73 3 f(g(t)) = f(t − 4) = 1 when 3 = 1 when 7 t. ≤ t − 4 ≤ The multipart rule for this composition can now be written dow... |
(x))) on the interval −2 x 10. ≤ ≤ (b) Your graphs should all intersect at the point (6,6). The value x = 6 is called a fixed point of the function f(x) since f(6) = 6; that is, 6 is fixed - it doesn’t move when f is applied to it. Give an explanation for why 6 is a fixed point for any function f(f(f(...f(x)...))). (c) Li... |
, g(x) = 4x2 + 2x + 1. (h) f(x) = −4, g(x) = 0. Problem 8.7. Let y = f(z) = √4 − z2 and z = g(x) = 2x + 3. Compute the composition y = f(g(x)). Find the largest possible domain of x-values so that the composition y = f(g(x)) is defined. Problem 8.8. Suppose you have a function y = f(x) such that the domain of f(x) is 1 ... |
remaining, tabulate your results, then sketch a graph as indicated in Figure 9.1. 1 fraction Viewing the input value as “time” and the output value as “fraction of product,” we could find a function y = f(t) modeling this data. Using this function, you can easily compute the fraction of reactants remaining at any time ... |
;” i.e., given x, we simply put it into a symbolic rule and out pops a new number f(x). This is all pretty mechanical and straightforward. 9.1.1 An Example Let’s schematically interpret what happens for the specific concrete example y = f(x) = 3x − 1, when x = −1, − 1 2, 1, 2: See Figure 9.3. 2, 0, 1 −1 − 1 2 in in in o... |
−1, we just have the first example above. For another example, suppose y = −0.8x + 2; then m = −0.8 and b = 2. In this case, the reverse process is −1.25(y − 2) = x. If we are given the value y = 11, we simply compute that x = −11.25; i.e., f(−11.25) = 11. 9.1.3 A Third Example The previous examples hide a subtle point... |
2 x = −√y − 1 3 3 (b) Two new reverse processes that are functions. Figure 9.6: What to do if a reverse process is not a function. Each of these “reverse processes” has a unique output; in other words, each of these “reverse processes” defines a function. So, given y = 3, there are TWO possible x values, √2, so that f(1... |
process” for a function: Important Fact 9.2.1. Given a number c, the x values such that f(x) = c can be found by finding the x-coordinates of the intersection points of the graphs of y = f(x) and y = c. Example 9.2.2. Graph y = f(x) = x2 and discuss the meaning of Fact 9.2.1 when c = 3, 1, 6. Solution. We graph y = x2 ... |
the other hand, if c then no input x value will lead to the output value c. For example, if f(x) = 1 and c = 1, then every real number can be input to produce an output of 1; if c = 2, then no input value of x will lead to an output of 2. y-axis f(x) = mx + b y-axis y = c x-axis the only input which leads to an output... |
-one function f(x) = x3. 9.3 Inverse Functions Let’s now come face to face with the problem of finding the “reverse process” for a given function y = f(x). It is important to keep in mind that the domain and range of the function will both play an important role in this whole development. For example, Figure 9.11 shows ... |
as a black box. What does it do? If we put in y in the input side, we should get out the x such that f(x) = y. in y f−1(y) out x such that f(x) = y Figure 9.12: A new function x = f−1(y). Now, let’s try to unravel something very special that is happening on a symbolic level. What would happen if we plugged f(a) into t... |
) for some x in the domain of f(x); i.e., we are using the fact that the domain of f−1 equals the range of f. The function f−1(y) takes the number f(x) and sends it to x, by Fact 9.3.2. So when f(x) is the input value, x becomes the output value. Conclude a point on the graph of f−1(y) looks like (f(x), x). It’s simila... |
’s try to see what this means graphically. Let’s set f(x) = x2, but only for non-negative x-values. That means that we want to erase the graph to the left of the y-axis (so remember - no negative x-values allowed). The graph would then look like Figure 9.15. +y y = x2 +x inverse function √y +x +y domain non-negative x ... |
every one-to-one function has an inverse. The domain of a function is the range of its inverse, and the range of a function is the domain of its inverse. The graph of a function and its inverse are mirror images of each other across the line y = x. 130 CHAPTER 9. INVERSE FUNCTIONS 9.6 Exercises Problem 9.1. Let f(x) =... |
the formula y = −2x2 + 120x. (a) Give a function h = f(x) relating the height h of the rocket above the sloping ground to its x-coordinate. (b) Find the maximum height of the rocket above the sloping ground. What is its x-coordinate when it is at its maximum height? (c) Clovis measures its height h of the rocket above... |
Make sure to specify the domain and compute the range too. Problem 9.9. A biochemical experiment involves combining together two protein extracts. Suppose a function φ(t) monitors the amount (nanograms) of extract A remaining at time t (nanoseconds). Assume you know these facts: 1. The function φ is invertible; i.e., ... |
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