Chapter 12

(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" 1 to 6 . Is \(C 1\) -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?

Let \(C(q)\) be the cost (in dollars) of producing \(q\) items. Translate the following equations into words. (a) \(C(300)=800\) (b) \(C^{-1}(1000)=500\) (c) \(C^{\prime}(200)=1.5\)

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) \(f(x)=6 x-3\) (b) \(f(x)=(x-3)^{3}\)

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. \(C(3 A)\) ii. \(C^{-1}(6)\) iii. \(C^{-1}(1)\) (b) In this particular situation, which of the following statements are true? i. \(C(3 A)=3 C(A)\) ii. \(C^{-1}(2 x)=2 C^{-1}(x)\) iii. \(C^{-1}\left(\frac{x}{2}\right)=\frac{C^{-1}(x)}{2}\) iv. \(C^{-1}(x+x)=C^{-1}(2 x)\) (c) Only one of the statements above is true for any invertible function \(C\). Which statement is this?

For each of the functions below, find \(f^{-1}(x)\). (a) \(f(x)=2-\frac{x+1}{x}\) (b) \(f(x)=\frac{x^{5}}{10}+7\)

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 point on the graph of \(f^{-1}(x)\). (a) \(f(x)=2 x+1\) and \(f^{-1}(x)\) (b) \(f(x)=x^{2}-2, x>0\) and \(f^{-1}(x)\) (c) \(f(x)=10^{x}\) and \(f^{-1}(x)\) (d) \(f(x)=2^{-x}\) and \(f^{-1}(x)\)

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. " \(A^{\prime \prime}\) and " \(B^{\prime \prime}\) 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) \(C^{-1}(2 B)\)

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 \(\frac{\Delta y}{\Delta x}\) 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 \(16 .\)

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.

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) \(D(C+5)=1.1 B\) (c) \(D^{-1}(B+10)\)

Let $$ f(x)=\frac{2 x-1}{3 x+4} $$ Find \(f^{-1}(x)\)

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; \(w \in(0,50]\). (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,\(1998 ; t \in[0,365]\). (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; \(w \in[0,2]\). (d) \(M(t)\) is the mileage on a car \(t\) days after it was purchased; \(t \in[0,365]\).

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. (a) \(R(B / 2)=C-80,000\) (b) \(R^{\prime}(30,000)=2.8\) (c) \(R^{-1}(2 C)\)

The function \(f\) is increasing and concave up on \((-\infty, \infty) \cdot f^{\prime}(x)\) is never zero. Denote by \(g(x)\) the inverse of \(f\). (a) What is the sign of \(g^{\prime}\) ? (b) What is the sign of \(g^{\prime \prime}\) ? (c) If \(f(3)=5\) and \(f^{\prime}(3)=10\), what is \(g^{\prime}(5) ?\)

Let \(f(x)=x^{3}+3 x^{2}+6 x+12\). (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 \((-\infty, \infty) ?)\) (b) Find three points that lie on the graph of \(f^{-1}(x)\). (Approximations are not adequate.) Explain your reasoning.

Let \(f(t)=5(1.1)^{6 t+2}+1\). (a) The point \(\left(\frac{1}{3}, 6\right)\) 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.

The functions in Problems 6 through 10 are 1 -to- \(1 .\) Find \(f^{-1}(x)\) and specify the domain of \(f^{-1}\). $$ f(x)=\frac{x}{x+3} $$

A ball is thrown straight up into the air. \(t\) seconds after it is released, its height is given by \(H(t)=-16 t^{2}+96 t\) 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 \(H(t) .\) Is it a 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?

The functions in Problems are 1 -to- \(1 .\) Find \(f^{-1}(x)\) and specify the domain of \(f^{-1}\) $$ f(x)=\frac{2}{3-x} $$

The functions in Problems are 1 -to- \(1 .\) Find \(f^{-1}(x)\) and specify the domain of \(f^{-1}\) $$ f(x)=\sqrt{x+3} $$

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.

The functions in Problems are 1 -to- \(1 .\) Find \(f^{-1}(x)\) and specify the domain of \(f^{-1}\) $$ f(x)=2 \sqrt{x-6} $$

The functions in Problems are 1 -to- \(1 .\) Find \(f^{-1}(x)\) and specify the domain of \(f^{-1}\) $$ f(x)=x^{3}+1 $$

For Problems 11 through 16, use the rst derivative to determine whether the function given is 1 -to- \(1 .\) If it is, nd its inverse function. $$ f(x)=x^{3}+2 x-3 $$

For Problems use the rst derivative to determine whether the function given is 1 -to- \(1 .\) If it is, nd its inverse function.. $$ f(x)=3 \cdot 2^{x} $$

For Problems use the rst derivative to determine whether the function given is 1 -to- \(1 .\) If it is, nd its inverse function.. $$ f(x)=5 \cdot 3^{-x} $$