Section 1
Inverse Functions
Find the inverse of the relation.$$\{(7,8),(-2,8),(3,-4),(8,-8)\}$$
Find the inverse of the relation.$$\{(0,1),(5,6),(-2,-4)\}$$
Find the inverse of the relation.$$\{(-1,-1),(-3,4)\}$$
Find the inverse of the relation.$$\{(-1,3),(2,5),(-3,5),(2,0)\}$$
Find an equation of the inverse relation.$y=4 x-5$
Find an equation of the inverse relation.$2 x^{2}+5 y^{2}=4$
Find an equation of the inverse relation.$x^{3} y=-5$
Find an equation of the inverse relation.$y=3 x^{2}-5 x+9$
Find an equation of the inverse relation.$$x=y^{2}-2 y$$
Find an equation of the inverse relation.$$x=\frac{1}{2} y+4$$
Graph the equation by substituting and plotting points. Then reflect the graph across the line $y=x$ to obtain the graph of its inverse.$x=y^{2}-3$
Graph the equation by substituting and plotting points. Then reflect the graph across the line $y=x$ to obtain the graph of its inverse.$y=x^{2}+1$
Graph the equation by substituting and plotting points. Then reflect the graph across the line $y=x$ to obtain the graph of its inverse.$y=3 x-2$
Graph the equation by substituting and plotting points. Then reflect the graph across the line $y=x$ to obtain the graph of its inverse.$x=-y+4$
Graph the equation by substituting and plotting points. Then reflect the graph across the line $y=x$ to obtain the graph of its inverse.$y=|x|$
Graph the equation by substituting and plotting points. Then reflect the graph across the line $y=x$ to obtain the graph of its inverse.$x+2=|y|$
Given the function $f,$ prove that $f$ is one-to-one using the definition of a one-to-one function on p. 390 .$f(x)=\frac{1}{3} x-6$
Given the function $f,$ prove that $f$ is one-to-one using the definition of a one-to-one function on p. 390 .$f(x)=4-2 x$
Given the function $f,$ prove that $f$ is one-to-one using the definition of a one-to-one function on p. 390 .$f(x)=x^{3}+\frac{1}{2}$
Given the function $f,$ prove that $f$ is one-to-one using the definition of a one-to-one function on p. 390 .$f(x)=\sqrt[3]{x}$
Given the function $g$, prove that $g$ is not one-to-one using the definition of a one-to-one function on p. 390 .$g(x)=1-x^{2}$
Given the function $g$, prove that $g$ is not one-to-one using the definition of a one-to-one function on p. 390 .$g(x)=3 x^{2}+1$
Given the function $g$, prove that $g$ is not one-to-one using the definition of a one-to-one function on p. 390 .$g(x)=x^{4}-x^{2}$
Given the function $g$, prove that $g$ is not one-to-one using the definition of a one-to-one function on p. 390 .$g(x)=\frac{1}{x^{6}}$
Using the horizontal-line test, determine whether the function is one-to-one.$f(x)=2.7^{x}$
Using the horizontal-line test, determine whether the function is one-to-one.$f(x)=2^{-x}$
Using the horizontal-line test, determine whether the function is one-to-one.$f(x)=4-x^{2}$
Using the horizontal-line test, determine whether the function is one-to-one.$f(x)=x^{3}-3 x+1$
Using the horizontal-line test, determine whether the function is one-to-one.$f(x)=\frac{8}{x^{2}-4}$
Using the horizontal-line test, determine whether the function is one-to-one.$f(x)=\sqrt{\frac{10}{4+x}}$
Using the horizontal-line test, determine whether the function is one-to-one.$f(x)=\sqrt[3]{x+2}-2$
Using the horizontal-line test, determine whether the function is one-to-one.$f(x)=\frac{8}{x}$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=5 x-8$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=3+4 x$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=1-x^{2}$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=|x|-2$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=|x+2|$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=-0.8$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=-\frac{4}{x}$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=\frac{2}{x+3}$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=\frac{2}{3}$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=\frac{1}{2} x^{2}+3$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=\sqrt{25-x^{2}}$
Graph the function and determine whether the function is one-to-one using the horizontal-line test.$f(x)=-x^{3}+2$
In Exercises $45-60,$ for each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=x+4$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=7-x$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=2 x-1$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=5 x+8$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=\frac{4}{x+7}$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=-\frac{3}{x}$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=\frac{x+4}{x-3}$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=\frac{5 x-3}{2 x+1}$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=x^{3}-1$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=(x+5)^{3}$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=x \sqrt{4-x^{2}}$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=2 x^{2}-x-1$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=5 x^{2}-2, x \geq 0$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=4 x^{2}+3, x \geq 0$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=\sqrt{x+1}$
For each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$f(x)=\sqrt[3]{x-8}$
Find the inverse by thinking about the operations of the function and then reversing, or undoing, them. Check your work algebraically.$$\begin{array}{lc}\text { FUNCTION } & \text { INVERSE } \\\hline f(x)=3 x & f^{-1}(x)=\end{array}$$
Find the inverse by thinking about the operations of the function and then reversing, or undoing, them. Check your work algebraically.$$\begin{array}{lc}\text { FUNCTION } & \text { INVERSE } \\\hline f(x)=\frac{1}{4} x+7& f^{-1}(x)=\end{array}$$
Find the inverse by thinking about the operations of the function and then reversing, or undoing, them. Check your work algebraically.$$\begin{array}{lc}\text { FUNCTION } & \text { INVERSE } \\\hline f(x)= -x& f^{-1}(x)=\end{array}$$
Find the inverse by thinking about the operations of the function and then reversing, or undoing, them. Check your work algebraically.$$\begin{array}{lc}\text { FUNCTION } & \text { INVERSE } \\\hline f(x)= \sqrt[3]{x}-5& f^{-1}(x)=\end{array}$$
Find the inverse by thinking about the operations of the function and then reversing, or undoing, them. Check your work algebraically.$$\begin{array}{lc}\text { FUNCTION } & \text { INVERSE } \\\hline f(x)= \sqrt[3]{x-5}& f^{-1}(x)=\end{array}$$
Find the inverse by thinking about the operations of the function and then reversing, or undoing, them. Check your work algebraically.$$\begin{array}{lc}\text { FUNCTION } & \text { INVERSE } \\\hline f(x)= x^{-1}& f^{-1}(x)=\end{array}$$
Each graph in Exercises $67-72$ is the graph of a one-to-one function $f$. Sketch the graph of the inverse function $f^{-1}$.
For the function $f$, use composition of functions to show that $f^{-1}$ is as given.$f(x)=\frac{7}{8} x, f^{-1}(x)=\frac{8}{7} x$
For the function $f$, use composition of functions to show that $f^{-1}$ is as given.$f(x)=\frac{x+5}{4}, f^{-1}(x)=4 x-5$
For the function $f$, use composition of functions to show that $f^{-1}$ is as given.$f(x)=\frac{1-x}{x}, f^{-1}(x)=\frac{1}{x+1}$
For the function $f$, use composition of functions to show that $f^{-1}$ is as given.$f(x)=\sqrt[3]{x+4}, f^{-1}(x)=x^{3}-4$
For the function $f$, use composition of functions to show that $f^{-1}$ is as given.$f(x)=\frac{2}{5} x+1, f^{-1}(x)=\frac{5 x-5}{2}$
For the function $f$, use composition of functions to show that $f^{-1}$ is as given.$f(x)=\frac{x+6}{3 x-4}, f^{-1}(x)=\frac{4 x+6}{3 x-1}$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.$f(x)=5 x-3$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.$f(x)=2-x$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.$f(x)=\frac{2}{x}$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.$f(x)=-\frac{3}{x+1}$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.$f(x)=\frac{1}{3} x^{3}-2$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.$f(x)=\sqrt[3]{x}-1$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.$f(x)=\frac{x+1}{x-3}$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.$f(x)=\frac{x-1}{x+2}$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.Find $f\left(f^{-1}(5)\right)$ and $f^{-1}(f(a))$ $f(x)=x^{3}-4$
Find the inverse of the given one-to-one function $f$ Give the domain and the range of fand of $f^{-1}$, and then graph both $\mathrm{fand} \mathrm{f}^{-1}$ on the same set of axes.Find $\left(f^{-1}(f(p))\right.$ and $f\left(f^{-1}(1253)\right):$$f(x)=\sqrt[5]{\frac{2 x-7}{3 x+4}}$
A function that will convert women's shoe sizes in the United States to those in Australia is$$s(x)=\frac{2 x-3}{2}$$(Source: OnlineConversion.com).a) Determine the women's shoe sizes in Australia that correspond to sizes $5,7 \frac{1}{2},$ and 8 in the United States.b) Find a formula for the inverse of the function.c) Use the inverse function to determine the women's shoe sizes in the United States that correspond to sizes $3,5 \frac{1}{2},$ and 7 in Australia.
A city swimming league determines that the cost per person of a group swim lesson is given by the formula$$C(x)=\frac{60+2 x}{x}$$where $x$ is the number of people in the group and $C(x)$ is in dollars. Find $C^{-1}(x)$ and explain what it represents.
The total amount of spending per year, in billions of dollars, on pets in the United States $x$ years after 2000 is given by the function$$P(x)=2.1782 x+25.3$$(Source: Animal Pet Products Manufacturing Association).a) Determine the total amount of spending per year on pets in 2005 and in 2010 .b) Find $P^{-1}(x)$ and explain what it represents.
The following formula can be used to convert Fahrenheit temperatures $x$ to Celsius temperatures $T(x)$$$T(x)=\frac{5}{9}(x-32)$$a) Find $T\left(-13^{\circ}\right)$ and $T\left(86^{\circ}\right)$.b) Find $T^{-1}(x)$ and explain what it represents.
Consider the following quadratic functions. Without graphing them, answer the questions below.a) $f(x)=2 x^{2}$b) $f(x)=-x^{2}$c) $f(x)=\frac{1}{4} x^{2}$d) $f(x)=-5 x^{2}+3$e) $f(x)=\frac{2}{3}(x-1)^{2}-3$f) $f(x)=-2(x+3)^{2}+1$g) $f(x)=(x-3)^{2}+1$h) $f(x)=-4(x+1)^{2}-3$Which functions have a maximum value?
Consider the following quadratic functions. Without graphing them, answer the questions below.a) $f(x)=2 x^{2}$b) $f(x)=-x^{2}$c) $f(x)=\frac{1}{4} x^{2}$d) $f(x)=-5 x^{2}+3$e) $f(x)=\frac{2}{3}(x-1)^{2}-3$f) $f(x)=-2(x+3)^{2}+1$g) $f(x)=(x-3)^{2}+1$h) $f(x)=-4(x+1)^{2}-3$Which graphs open up?
Consider the following quadratic functions. Without graphing them, answer the questions below.a) $f(x)=2 x^{2}$b) $f(x)=-x^{2}$c) $f(x)=\frac{1}{4} x^{2}$d) $f(x)=-5 x^{2}+3$e) $f(x)=\frac{2}{3}(x-1)^{2}-3$f) $f(x)=-2(x+3)^{2}+1$g) $f(x)=(x-3)^{2}+1$h) $f(x)=-4(x+1)^{2}-3$Consider (a) and (c). Which graph is narrower?
Consider the following quadratic functions. Without graphing them, answer the questions below.a) $f(x)=2 x^{2}$b) $f(x)=-x^{2}$c) $f(x)=\frac{1}{4} x^{2}$d) $f(x)=-5 x^{2}+3$e) $f(x)=\frac{2}{3}(x-1)^{2}-3$f) $f(x)=-2(x+3)^{2}+1$g) $f(x)=(x-3)^{2}+1$h) $f(x)=-4(x+1)^{2}-3$Consider (d) and (e). Which graph is narrower?
Consider the following quadratic functions. Without graphing them, answer the questions below.a) $f(x)=2 x^{2}$b) $f(x)=-x^{2}$c) $f(x)=\frac{1}{4} x^{2}$d) $f(x)=-5 x^{2}+3$e) $f(x)=\frac{2}{3}(x-1)^{2}-3$f) $f(x)=-2(x+3)^{2}+1$g) $f(x)=(x-3)^{2}+1$h) $f(x)=-4(x+1)^{2}-3$Which graph has vertex (-3,1)$?$
Consider the following quadratic functions. Without graphing them, answer the questions below.a) $f(x)=2 x^{2}$b) $f(x)=-x^{2}$c) $f(x)=\frac{1}{4} x^{2}$d) $f(x)=-5 x^{2}+3$e) $f(x)=\frac{2}{3}(x-1)^{2}-3$f) $f(x)=-2(x+3)^{2}+1$g) $f(x)=(x-3)^{2}+1$h) $f(x)=-4(x+1)^{2}-3$For which is the line of symmetry $x=0 ?$
The function $f(x)=x^{2}-3$ is not one-to-one. Restrict the domain of $f$ so that its inverse is a function. Find the inverse and state the restriction on the domain of the inverse.
Consider the function $f$ given by$$f(x)=\left\{\begin{array}{ll}x^{3}+2, & \text { for } x \leq-1 \\x^{2}, & \text { for }-1<x<1 \\x+1, & \text { for } x \geq 1\end{array}\right.$$Does $f$ have an inverse that is a function? Why or why not?
Find three examples of functions that are their own inverses; that is, $f=f^{-1}$.
Given the function $f(x)=a x+b, a \neq 0,$ find the values of $a$ and $b$ for which $f^{-1}(x)=f(x)$.