PART IV Inverse Functions: A Case Study of Exponential and Logarithmic Functions 12 CHAPTER Inverse Functions: Can What Is Done Be Undone? What’s done cannot be undone. —MacBeth, Act V, scene 1 Now mark me, how I will undo myself. —Richard II, Act IV, scene 1 12.1 WHAT DOES IT MEAN FOR F AND G TO BE INVERSE FUNCTIONS? Some actions can be undone; other actions, once taken, can never be undone. If we think of a function as an action on an input variable to produce an output, we can make a similar observation. Let’s begin by looking at functions whose actions can be undone. If a function acts on its input by adding 3, the action can be undone by subtracting 3. If a function acts by doubling its input, the action can be undone by halving. If f is a function whose action can be undone, we refer to the function that undoes the action of f as its inverse function and denote it by f −1 . We read f −1 as “f inverse.” CAUTION This notation, while quite standard, can be very misleading. “f inverse” is not 1 f(x) . The reciprocal of f(x)is written 1 f(x) or [f(x)] −1 . 421 422 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? ◆ EXAMPLE 12.1 Below are some simple functions together with their inverse functions. Function Action done Action undone Inverse function i. f(x)=x+3 Add 3 to input Subtract 3 from input f −1 (x) = x − 3 ii. g(x) = 2x Double input Halve input g −1 (x) = x/2 iii. h(x) = x 3 Cube the input Take the cube root h −1 (x) = x 1/3 Notice again that the inverse of f is not the reciprocal 1 of f . ◆ What exactly do we mean when we say “the inverse function of f undoes the action of f ”?Iff assigns to the input “a” the output “b,” then its inverse, f −1 , assigns to the input “b” the output “a.” That is, if f(a)=b, then f −1 (b) = a. Equivalently, for every point (a, b) on the graph of f , the point (b, a) lies on the graph of f −1 . input of f output of f –1 f –1 f y 1 y 2 x 2 x 1 x 3 y 3 a f → b f −1 → a If f and f −1 are inverse functions, f −1 undoes the action of f and vice versa,so a f →b f −1 → a and b f −1 → a f → b input of f a b f(a) = b input of f –1 output of f –1 f –1 (b) = a output of f Figure 12.1 Look at Example 12.1(i), where f(x)=x+3and f −1 (x) = x − 3. 4 f → 7 f −1 → 4 and 7 f −1 → 4 f → 7 5 f −1 → 2 f → 5 and 2 f → 5 f −1 → 2 More generally, imagine sending x down an assembly line. The first machine on the line, let’s call it f , acts on x. Its output, f(x),ispassed along to the second machine, f −1 , which undoes the work of the first. Therefore, the final output is x. x f → f(x) f −1 → x and similarly x f −1 → f −1 (x) f → x. 1 Do not confuse inverse functions with the statement “A is inversely proportional to B, A = k B .” 12.1 What Does It Mean for f and g to Be Inverse Functions? 423 We can express this more succinctly using the composition of functions. 2 f −1 (f (x)) = x and f(f −1 (x)) = x Definition The functions f and g are inverse functions if, for all x in the domain of f , g(f (x)) = x and for all x in the domain of g, f(g(x))=x. EXERCISE 12.1 Verify that the pairs of functions in Example 12.1 actually are inverse functions. We’ll do parts (i) and (ii) below. Part (iii) is left as an exercise. i. Verify that f(x)=x+3and f −1 (x) = x − 3 are inverse functions. x f → (x + 3) f −1 → (x + 3) − 3 = x; f −1 (f (x)) = f −1 (x + 3) = (x + 3) − 3 = x. x f −1 → (x − 3) f → (x − 3) + 3 = x; f(f −1 (x)) = f(x −3)=(x − 3) + 3 = x. ii. Verify that g(x) = 2x and g −1 (x) = x 2 are inverse functions. x g → 2x g −1 → 2x 2 = x; g −1 (g(x)) = g −1 (2x) = 2x 2 = x. x g −1 → x 2 g → 2  x 2  = x; g(g −1 (x)) = g  x 2  = 2  x 2  = x. Definition A function is said to be invertible if it has an inverse function. Let’s review the characteristic of a function that makes it invertible, able to be undone. A function is an input/output relationship such that each input corresponds to a single output. To fi nd the inverse of a function requires that we are able to begin with an output and trace it back to the corresponding input. Therefore no two inputs may share the same output. In other words, the function must be 1-to-1; there must be a 1-to-1 correspondence between inputs and outputs. To illustrate this we’ll reconstruct the soda machine example from Chapter 1. Both machine A and machine B can be modeled by functions, but only machine A is 1-to-1. 2 Recall that f(g(x)) means find g(x) and use g(x) as the input of f . In other words, do f to g(x). Therefore f(f −1 (x)) means do f −1 first and apply f to the output of f −1 . For a review of composition of functions, refer to Chapter 3. 424 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? Machine A Machine B Button # Output Button # Output 1 Coke 1 Coke 2 Diet Coke 2 Coke 3 Sprite 3 Diet Coke 4 Orange Crush 4 Diet Coke 5 Ginger Ale 5 Coke 6 Coke Soda machine B has six input buttons, but can give outputs of only Coke and Diet Coke; the function that models this machine is not 1-to-1. Knowing that the machine gave an output of Coke does not allow us to determine precisely which button was pressed. On the other hand, soda machine A, with five selection buttons, each one corresponding to a different type of soda, is modeled by a 1-to-1 function. To be invertible a function must be 1-to-1 because the inverse of f must map each output of f to the unique corresponding input; conversely, if f is 1-to-1, then f is invertible. Let f be a function modeling machine A. f and its inverse function f −1 are shown below. 1 f → Coke f −1 → 1 2 −→ Diet Coke −→ 2 3  −→ Sprite −→ 3 4  −→ Orange Crush −→ 4 5  −→ Ginger Ale −→ 5 Do we care whether a given function is invertible? It depends on the situation. For the soda machines above, it’s probably not very important to be able to tell what button was pressed based on the type of soda that came out of the machine. It’s merely important that the relationship between button and soda output be a function so that the input uniquely determines the output. On the other hand, recall the bottle calibration problem from the beginning of the course. The entire exercise of calibrating a bottle is worthless unless this calibrated bottle can be used for measuring. We want to be able to pour water into the bottle and determine its volume based on the height of the water. It’s important that the calibration function is invertible. The calibration function, C, takes volume as input and assigns height as output. For instance, one liter of water might fill the bottle to a height of 10 cm, in which case C(1) = 10. Once calibrated, the bottle can be used as a measuring device precisely because the calibration function is 1-to-1. C −1 turns the calibration procedure around; the height becomes the input and the volume the output. Because the function is 1-to-1 we know that a height of 10 cm corresponds to a volume of 1 liter, C −1 (10) = 1. volume C −→ height height C −1 −→ volume ◆ EXAMPLE 12.2 Let’s consider a particular bottle and its calibration function C. From the information about C given on the left in the following table we can construct a corresponding table for C −1 . 12.1 What Does It Mean for f and g to Be Inverse Functions? 425 v C(v) = hhC −1 (h) = v (in liters) (in cm) ⇒ corresponding table (in cm) (in liters) 0.25 4 for C −1 4 0.25 0.5 7 7 0.5 0.75 9 9 0.75 110 101 The graph of C is given in Figure 12.2(a) below. Since the function C −1 reverses the input and output of C, we can graph C −1 by reversing the coordinates of the points on the graph of C. height (cm) volume (liters) 4 8 12 .25 .5 .75 1 (.25, 4) (.75, 9) (.5, 7) h = C(v) height volume .25 .5 .75 4812 (4, .25) (9, .75) (7, .5) V = C –1 (h) (a) (b) Figure 12.2 ◆ The Relationship Between the Graph of a Function and the Graph of Its Inverse We have established that a function is invertible if and only if the function is 1-to-1. 3 How is this criterion reflected in the graph of y = f(x)?Iff is invertible, then each y-value in the range must correspond to exactly one x-value in the domain. f x y 0 f x (i) f is not invertible on [a, b]. There are y-values corresponding to more than 1 x-value. (ii) f is invertible on [a, b]. There is exactly 1 x-value corresponding to each y-value. abab Figure 12.3 This gives us a graphical criterion for determining if a function is invertible. 3 Recall that A if and only if B means that A and B are equivalent statements. If a function is invertible, then it is 1-to-1; if a function is 1-to-1, then it is invertible. 426 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? The Horizontal Line Test. A function f is invertible if and only if every horizontal line intersecting the graph of f intersects it in exactly one point. The horizontal line test is the reflection of the vertical line test about the line y = x. The vertical line test checks that there is at most one y for every x; that is, there is at most one output for each input. The horizontal line test checks that there is at most one x for every y; that is, there is at most one input for each output. Together, they check that the relationship is 1-to-1. Consequence: If a continuous function f is invertible, then f must be either always increasing or always decreasing. A function with a turning point is not invertible; it cannot pass the horizontal line test. We can ascertain this information by calculating f  .Iff is continuous on an open interval and f  is either always positive or always negative, then f has no turning points and therefore passes the horizontal line test and is invertible; if f is continuous and f  changes signs, then f has a turning point and therefore fails the horizontal line test and is therefore not invertible. 4 f x f is not invertible f x f is invertible Figure 12.4 As illustrated in Example 12.2, the graph of f −1 is obtained by interchanging the coordinates of the points on the graph of f .If(a, b) lies on f , then (b, a) lies on the graph of f −1 . This is equivalent to reflecting the graph of y = f(x)overthe line y = x as shown in the examples below. The graphs in Figure 12.5 correspond to the functions and inverse functions given in Example 12.1. x y y = x f f(x) = x + 3 f –1 (x) = x – 3 f –1 3 3–3 –3 (i) xx yy y = x g g(x) = 2x g –1 (x) = x/2 g –1 h –1 h y = x (ii) h(x) = x 3 h –1 (x) = x (iii) 1 3 Figure 12.5 4 Note that it is not sufficient to check only if f  ever equals 0. For example, f(x)=x 3 is invertible since it has no turning points, although f  (0) = 0. 12.1 What Does It Mean for f and g to Be Inverse Functions? 427 PROBLEMS FOR SECTION 12.1 1. (a) Let S be the function that assigns to each living person a social security number. Is S 1-to-1? Is it invertible? (b) Let C be the counting function that allows a collection of 30 people to be put in six groups of five people each by “counting off” 1to6.IsC1-to-1? Is it invertible? (c) Let A be the altitude function that assigns to each point in the White Mountains its altitude. Is A 1-to-1? 2. The identity function I is the function whose input equals its output: I(x)=x.If functions f and g have the property that f(g(x))=I(x) and g(f (x)) = I(x), then f and g are inverse functions. For each function below, find the inverse function g(x) and verify that f(g(x))=I(x)and g(f (x)) = I(x). (a) f(x)=6x −3 (b) f(x)=(x − 3) 3 3. On the same set of axes, sketch the graphs of the following pairs of functions. In parts (a) and (b) find an expression for f −1 (x). The graphs of f and f −1 (x) are mirror images over the line y = x since the roles of input and output are switched to obtain the inverse function. In other words, if (1, 5) is a point on the graph of f , then (5, 1) is a point on the graph of f −1 (x). (a) f(x)=2x +1and f −1 (x) (b) f(x)=x 2 −2, x>0and f −1 (x) (c) f(x)=10 x and f −1 (x) (d) f(x)=2 −x and f −1 (x) 4. Suppose f(v)is a calibration function for a bucket. f takes volumes (in liters) as inputs and gives heights (in inches) as outputs. Suppose f(1)=4. (a) What is f −1 (4)? (b) What is the meaning of f −1 (4) in physical terms? (c) Is f −1 (4) greater than f −1 (1)? Explain in terms of the physical situation. 5. Which of the following functions are invertible on the domain given? Explain. (a) P(w)is the price of mailing a package weighing w ounces; w ∈ (0, 50]. (b) T(t) is the temperature at the top of the Prudential Center in Boston at time t, t measured in days, where t = 0 is February 1, 1998; t ∈ [0, 365]. (c) C(w) is the cost of w pounds of ground coffee at a particular shop where coffee is sold by weight at a fixed price per pound; w ∈ [0, 2]. (d) M(t) is the mileage on a car t days after it was purchased; t ∈ [0, 365]. 6. Let f(x)=x 3 +3x 2 +6x+12. (a) Make a convincing argument that f(x) is invertible. (It is not adequate to say it looks 1-to-1 on a calculator. How can you be absolutely sure it is 1-to-1 on (−∞, ∞)?) (b) Find three points that lie on the graph of f −1 (x). (Approximations are not ade- quate.) Explain your reasoning. 428 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? 7. For each of the functions graphed below, determine whether or not the function is invertible. If it is not, restrict the domain to make it invertible. Then sketch f −1 , labeling any asymptotes and labeling two points on the graph of f −1 . (a) (b) (c) yyy xx x = x (2, 4) 1 f(x) f(x) –ππ –π ( / 2 –π / 2 x = π / 2 , –1) π ( / 2 , 1) π ( / 4 , 1) 8. The graphs of f(t),g(t), and h(t) are given below. y t g(t) ππ –π –π–π h(t) y t x = x = 2 –π 2 π 2 y t f(t) π –π –π 2 π 2 π 2 True or False? (a) The function f(t),restricted to the domain [0, π], is invertible. (b) The function g(t), restricted to the domain [0, π], is invertible. (c) The function h(t), restricted to the domain (− π 2 , π 2 ), is invertible. (d) The function f(t),restricted to the domain [− π 2 , π 2 ], is invertible. 9. Which of the following functions are invertible? (a) The function that assigns to each current senator the state he or she represents. (b) The function T(t)that gives the temperature in Moab at time t. (c) The function C(d), whose domain is the set of all performances of Broadway’s A Chorus Line, and whose output is the cumulative number of people who have seen this show on Broadway. (d) The function L(d), whose domain is the set of all performances of Broadway’s The Lion King, and whose output is the number of people seeing this Broadway show on the designated date. 12.2 Finding the Inverse of a Function 429 12.2 FINDING THE INVERSE OF A FUNCTION If an invertible function is given by a table of values, the inverse function is constructed by interchanging the input and output columns. If an invertible function is presented graphically, the graph of f −1 is obtained by reflecting the graph of f over the line y = x. Suppose a function is given analytically. The subject of this section is how to arrive at an expression for its inverse function. ◆ EXAMPLE 12.3 Suppose f is the function that doubles its input and then adds 3: f(x)=2x+3. What is its inverse function? SOLUTION To undo the function f , do we subtract 3 and then divide by 2, or do we first divide by 2 and then subtract 3; is f −1 (x) = x−3 2 or is f −1 (x) = x 2 − 3? We could try each out, since we know that f −1 (f (x)) should be x, or we can think about it in the following way. Analogy. When getting dressed, you first put on your socks and then put on your shoes. To undo the process, you must first remove your shoes, then your socks. The last thing you did is the first thing you undo. Accordingly, to undo f(x),wefirst subtract 3 and then divide the result by 2: f −1 (x) = x−3 2 . x multiply by 2 add 3 2x 2x + 3 divide by 2 subtract 3 f –1 (2x + 3) = x f(x) = 2 x + 3 Check: f −1 (f (x)) = f −1 (2x + 3) = (2x+3)−3 2 = 2x 2 = x. f(f −1 (x)) = f  x−3 2  = 2  x−3 2  + 3 = (x − 3) + 3 = x. ◆ The three functions from Example 12.1 and the function from the example above had simple enough formulas that we could guess how to “undo” them in order to find formulas for their inverses. As this is not always the case, we need an analytic method for finding a formula for the inverse function. ◆ EXAMPLE 12.4 Let f(x)= x+1 x−2 . Find a formula for f −1 (x). SOLUTION In this example it is not very easy to figure out how to undo the action of f ,sowe’ll use a different approach: f −1 reverses the output and input of f . Let’s denote the output of f by y and the input by x. f is given by y = x+1 x−2 . We interchange the input and output (x and y) to obtain f −1 . 430 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? x = y + 1 y − 2 , where x is the input of f −1 and y is the output of f −1 . We solve for y to get the output of f −1 in terms of the input. x = y + 1 y − 2 Get y out of the denominator by multiplying both sides by (y − 2). x(y − 2) = y + 1 Multiply out. xy − 2x = y + 1 Gather the y’s together. xy − y = 2x + 1 Factor out y. y(x − 1) = 2x + 1 Solve for y. y = 2x + 1 x − 1 So f −1 (x) = 2x+1 x−1 . ◆ To summarize, we find a formula for f −1 by exploiting the fact that the inverse of f reverses the roles of input and output. Write y = f(x),and then interchange the variables x and y. Solve for y in terms of x.We’ll obtain y = f −1 (x). ◆ EXAMPLE 12.5 Let f(x)=4x 3 +2. Find f −1 (x) if f is invertible. SOLUTION We know that f is invertible because f  (x) = 12x 2 is positive for all x = 0. This indicates that f is a cubic polynomial with no turning points, so it’s 1-to-1. Set y = 4x 3 + 2 and then interchange the roles of x and y to find the inverse relationship. x = 4y 3 + 2 4y 3 = x − 2 y 3 = x − 2 4 y =  x − 2 4  1/3 f −1 (x) =  x − 2 4  1/3 . MEAN FOR F AND G TO BE INVERSE FUNCTIONS? Some actions can be undone; other actions, once taken, can never be undone. If we think of a function as an action on an input variable to produce an. interchange the input and output (x and y) to obtain f −1 . 430 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? x = y + 1 y − 2 , where x is the input of f −1 and y is the output of f −1 points and therefore passes the horizontal line test and is invertible; if f is continuous and f  changes signs, then f has a turning point and therefore fails the horizontal line test and is