12.2 Finding the Inverse of a Function 431 x y f(x) f –1 (x) 2 2 Figure 12.6 This makes sense: The function f cubes its input, multiplies the result by 4, and then adds 2. To undo this, we must subtract 2, divide the result by 4, and then take the cube root. See the diagram below for clarification. x cube multiply by 4 x 3 4x 3 cube root divide by 4 f –1 f subtract 2 4x 3 +2 add 2 ◆ ◆ EXAMPLE 12.6 Let g(x) = x 2 . Find g −1 (x) if g is invertible. At first glance, it may seem that if g is the squaring function, its inverse must be the square root function. But we must be careful. g is not invertible because it is not 1-to-1. The problem is that given any positive output, say 4, it is impossible to determine uniquely the corresponding input. The input corresponding to 4 could be 2 or −2. 432 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? x y g(x) g is NOT invertible Figure 12.7 Had we not noticed this and proceeded to look for the inverse analytically, we would soon realize that there was a problem. Set y = x 2 and then interchange the roles of x and y to obtain an inverse relationship. x = y 2 Solve for y. y =± √ xyis not a function of x. What we can do is restrict the domain of g to make g 1-to-1. If we restrict the domain to [0, ∞), then g −1 (x) = √ x. If we restrict the domain to (−∞, 0], then g −1 (x) =− √ x. yy x x g(x) = x 2 g(x) = x 2 g –1 (x) = √x g –1 (x) = – √x (i) g(x) on [0, ∞) g –1 (x) = √x (ii) g(x) on (–∞, 0] g –1 (x) = – √x Figure 12.8 ◆ Observation If a function is not 1-to-1 on its natural domain it is possible to restrict the domain in order to make the function invertible. Note that the domain of f is the range of f −1 and the range of f is the domain of f −1 . PROBLEMS FOR SECTION 12.2 1. For each of the functions f , g, and h below, do the following. (a) Sketch the function and determine whether it is invertible. (b) If the function is invertible, sketch the inverse function on the same set of axes as the function and find a formula for the inverse function. 12.2 Finding the Inverse of a Function 433 (c) Identify the domain and range of the function and its inverse. i. f(x)= √ x − 1 ii. g(x) = 1 x iii. h(x) = x 3 3 + 1 2. For each of the functions below, find f −1 (x). (a) f(x)=2− x +1 x (b) f(x)= x 5 10 + 7 3. Suppose f is an invertible function. (a) If f is increasing, is f −1 increasing, decreasing, or is there not enough information to determine? (b) If f is decreasing, is f −1 increasing, decreasing, or is there not enough information to determine? (c) Suppose f is increasing and concave up. Is f −1 concave up or concave down? (Hint: Let y = f(x).What happens to the ratio y x as x increases? How does this translate into information about the inverse function? Check your conclusion with a concrete example.) We will be able to work this out analytically by Chapter 16. 4. Let f(x)= 2x −1 3x + 4 . Find f −1 (x). 5. The function f is increasing and concave up on (−∞, ∞). f (x) is never zero. Denote by g(x) the inverse of f . (a) What is the sign of g ? (b) What is the sign of g ? (c) If f(3)=5and f (3) = 10, what is g (5)? The functions in Problems 6 through 10 are 1-to-1. Find f −1 (x) and specify the domain of f −1 . 6. f(x)= x x+3 7. f(x)= 2 3−x 8. f(x)= √ x + 3 9. f(x)=2 √ x − 6 10. f(x)=x 3 +1 ForProblems 11 through 16, use the first derivative to determine whether the function given is 1-to-1. If it is, find its inverse function. 11. f(x)=x 3 +2x −3 12. f(x)=x 3 −2x +3 434 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? 13. f(x)=|x−3| 14. f(x)=x 2 +2x −1 15. f(x)=3·2 x 16. f(x)=5·3 −x 12.3 INTERPRETING THE MEANING OF INVERSE FUNCTIONS In order to make sense of the information given by an inverse function in some real-world context it is helpful to clarify in words the input and output of both the function and its inverse. We illustrate this in the example below. ◆ EXAMPLE 12.7 Let P(t)be the amount of money in a bank account at time t, where t is measured in years and t = 0 represents January 1, 1998. Suppose that P 0 dollars are originally deposited in the account. Interpret the following expressions in words. (a) P(2) (b) P −1 (5000) (c) P −1 (2P 0 ) (d) P −1 (2P 0 + 10) (e) P −1 (2P 0 ) + 10 SOLUTION Begin by firmly establishing the input and output of P and P −1 . P P –1 time dollars P (time) = dollars, so P −1 (dollars) = time (a) P(2)is the number of dollars in the account at t = 2 (January 1, 2000). (b) P −1 (5000) is the time when the balance will be $5000. (c) P −1 (2P 0 ) is the time when the balance will be twice P 0 , i.e., when the balance will have doubled from its original amount. (d) P −1 (2P 0 + 10) is the time when the balance will be $10 more than twice the original balance. (e) P −1 (2P 0 ) + 10 is the time ten years after the balance is twice P 0 , i.e., ten years after the balance has doubled. ◆ PROBLEMS FOR SECTION 12.3 1. Let C(q)be the cost (in dollars) of producing q items. Translate the following equations into words. 12.3 Interpreting the Meaning of Inverse Functions 435 (a) C(300) = 800 (b) C −1 (1000) = 500 (c) C (200) = 1.5 2. Apricots are sold by weight. In other words, the price is proportional to the weight. Let C(w) be the cost of w pounds of apricots. Suppose that A pounds of apricots cost $3. (a) Describe in words the practical meaning of each of the following and then evaluate the expression. (When evaluating, use the fact that price is proportional to weight. Your answers should be either a number or an expression in terms of A.) i. C(3A) ii. C −1 (6) iii. C −1 (1) (b) In this particular situation, which of the following statements are true? i. C(3A) = 3C(A) ii. C −1 (2x) = 2C −1 (x) iii. C −1 ( x 2 ) = C −1 (x) 2 iv. C −1 (x + x) = C −1 (2x) (c) Only one of the statements above is true for any invertible function C. Which statement is this? 3. Let C(q) be the cost of producing q items. Suppose that right now A items have been produced at a cost of $B. Interpret the following expressions in words. “A” and “B” should not appear in your answers; use words instead. (a) C(400) (b) C −1 (3000) (c) C −1 (B + 100) (d) C(A + 10) (e) C −1 (2B) 4. A typist’s daily wages are determined by the number of words per minute he averages on his shift. Let D(w) be his daily earnings (in dollars) as a function of w, the average number of words per minute he types. Suppose that yesterday he was paid $B for averaging C words per minute. Interpret each of the following equations or expressions in words. Your answer should be expressed in terms of pay and words per minute. (a) D −1 (70) = 50 (b) D(C + 5) = 1.1B (c) D −1 (B + 10) 5. Let R(d) be a function that models a company’s annual revenue (the amount of money they receive from customers) in dollars as a function of the number of dollars they spend that year on advertising. Suppose that last year they spent $B on advertising and took in a total revenue of $C. Interpret each of the following equations or expressions. Your answers should not contain $C or $B, but words instead. 436 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? (a) R(B/2) = C − 80,000 (b) R (30,000) = 2.8 (c) R −1 (2C) 6. Let f(t)=5(1.1) 6t+2 + 1. (a) The point ( 1 3 ,6)lies on the graph of f . What is f −1 (6)? (b) Find a formula for f −1 (t). (c) Use your formula to find f −1 (6). Does your answer agree with your answer to part (a)? (d) If f(t) models the number of pounds of garbage in a garbage dump t days after the dump has officially opened, interpret f −1 (30) in words. 7. A ball is thrown straight up into the air. t seconds after it is released, its height is given by H(t)=−16t 2 + 96t feet. (a) Sketch a graph of H(t). (b) What is the domain of H(t)? The range? (c) What is the ball’s maximum height? When does it attain this height? (d) Sketch the inverse relation for H(t).Isita function? Explain. (e) How can you restrict the domain of H(t)so that it will have an inverse? (f) Having restricted the domain so that H(t) is invertible, evaluate H −1 (80). What is its practical meaning? Exploratory Problems for Chapter 12 437 Exploratory Problems for Chapter 12 Thinking About the Derivatives of Inverse Functions 1. Let f(x)=x 2 with the domain of f restricted to x ≥ 0. Then f −1 (x) = √ x. (a) Compare the derivative of x 2 at (3, 9) with the derivative of √ x at (9, 3). Accompany your answer by a sketch. (b) Compute the derivative of x 2 at (a, b). (c) Compute the derivative of √ x at (b, a). (d) Compare your answers to the previous two questions by either expressing both in terms of a or expressing both in terms of b. How are they related? 2. The function e x is invertible. Denote its inverse function by g(x). (a) On the same set of axes, graph e x and its inverse function g(x). What is the domain of g? The range of g? (b) The derivative of e x at (0, 1) is 1. What do you think the derivative of the inverse function is at (1, 0)? (c) What can you say about the sign of g and of g ? 13 CHAPTER Logarithmic Functions 13.1 THE LOGARITHMIC FUNCTION DEFINED Introductory Example ◆ EXAMPLE 13.1 A large lake has been serving as a reservoir for its nearby towns. Over the years, industries on the shore of the lake have contributed to the pollution of the lake. An awareness of the problem has caused community members to ban further pollution. Due to a combination of runoff and natural processes, the amount of pollutants in the lake is expected to decrease at a rate proportional to pollution levels. The number of grams of pollutant in the lake is now 1200; t years from now the number of grams is expected to be given by 1200(10) −t/8 .If the water is deemed safe to drink only when the pollutant level has dropped to 400 grams, for how many years will the towns need to find an alternative source of drinking water? SOLUTION We must find t such that 400 = 1200(10) −t/8 . This is equivalent to solving 400 1200 = (10) −t/8 , or 1 3 = (10) −t/8 .Wecan approximate the solution using a graphing calculator. One approach is to look for the root of 1 3 − (10) −t/8 . Another is to look for the point of intersection of y = 1200(10) −t/8 and y = 400. But suppose we would like an exact answer; we want to solve the equation analytically for t. We could simplify somewhat by converting 1 3 = (10) −t/8 to 1 3 = (10 t ) −1/8 and raising both sides of this equation to the (−8) to get 1 3 −8 = 10 t . We know 1 3 −8 = ( 3 ) −1 −8 = (3) 8 = 6561, so we must solve the equation 10 t = 6561. t is the number we must raise 10 to in order to get 6561. Since 10 3 = 1000 and 10 4 = 10,000, we can be sure that t is a number between 3 and 4. Again we could revert to our calculator to get better and better estimates of the value of t.However,ifwecanfind the inverse of the 439 440 CHAPTER 13 Logarithmic Functions function 10 t , then we can find the exact solution to the equation 10 t = 6561. If f(t)=10 t , then t = f −1 (6561). The Inverse of f(x)=10 x We know the function f(x)=10 x is invertible because it is 1-to-1. The inverse function, f −1 , is obtained by interchanging the input and output of f ; the graph of f −1 can be drawn by reflecting the graph of 10 x over the line y = x. x y y = x f –1 (x) f (x) = 10 x 1 1 Figure 13.1 Suppose we go about looking for a formula for f −1 in the usual way, by interchanging the roles of x and y and solving for y. We write x = 10 y . What is y? y is the number we must raise 10 to in order to get x.Wedon’thaveanalgebraic formula for this, but this function is quite useful, so we give it a name. Definition log 10 x is the number we must raise 10 to in order to obtain x. log 10 x is often written log x . We read log 10 x as “log base 10 of x.” y = log 10 x is equivalent to 10 y = x. The domain of f is (−∞, ∞) and the range is (0, ∞). Therefore the domain of log x is (0, ∞) and its range is (−∞, ∞). Note then that log x is defined only for x>0. By examining the graph in Figure 13.1, we see that the graph of log x is increasing and concave down; it is increasing without bound, but it is increasing very slowly. lim x→0 + log 10 x =−∞ lim x→∞ log 10 x =+∞ Note that although we now have a nice compact way of expressing the inverse function of 10 x , it might seem that all we have really accomplished so far is the introduction of a shorthand for writing “the number we must raise 10 to in order to obtain x.” But there isadefinite perk. A calculator will give a numerical estimate of the logarithm up to 10 . 10 to in order to get 6561. Since 10 3 = 1000 and 10 4 = 10,000, we can be sure that t is a number between 3 and 4. Again we could revert to our calculator to get better and better estimates of. the shore of the lake have contributed to the pollution of the lake. An awareness of the problem has caused community members to ban further pollution. Due to a combination of runoff and natural. calculator. One approach is to look for the root of 1 3 − (10) −t/8 . Another is to look for the point of intersection of y = 1200(10) −t/8 and y = 400. But suppose we would like an exact answer;