Section 1
Inverse Functions
The table shows the number of registered passenger cars in the United States for the years $2012-2016$ $$\begin{array}{c|c}\text { Year } & \begin{array}{c}\text { Registered Passenger Cars } \\\text { (in thousands) }\end{array} \\\hline 2012 & 111,290 \\\hline 2013 & 113,676 \\\hline 2014 & 113,899 \\\hline 2015 & 112,864 \\\hline 2016 & 112,961 \\\hline\end{array}$$
The table gives the number of representatives currently in Congress from each of five New England states.
$$\text { For a function to have an inverse, it must be }$$
If two functions $f$ and $g$ are inverses, then $(f \circ g)(x)=$ ____and____$$=x$$
The domain of $f$ is equal to the ___of $f^{-1},$ and the range of $f$ is equal to the ___of $f^{-1}$
If the point $(a, b)$ lies on the graph of $f,$ and $f$ has an inverse, then the point ____lies on the graph of $f^{-1}$.
If $f(x)=x^{3},$ then $f^{-1}(x)=$_____
If a function $f$ has an inverse, then the graph of $f^{-1}$ may be obtained by reflecting the graph of $f$ across the line with equation _____
If a function $f$ has an inverse and $f(-3)=6,$ then $f^{-1}(6)=$_____
If $f(-4)=16$ and $f(4)=16,$ then $f __________\underbrace{(\text { does } / \text { does not })}$ have an inverse because________
Determine whether each function graphed or defined is one-to-one.
Determine whether each function graphed or defined is one-to-one. $y=2 x-8$
Determine whether each function graphed or defined is one-to-one $y=4 x+20$
Determine whether each function graphed or defined is one-to-one. $y=\sqrt{36-x^{2}}$
Determine whether each function graphed or defined is one-to-one. $y=-\sqrt{100-x^{2}}$
Determine whether each function graphed or defined is one-to-one $y=2 x^{3}-1$
Determine whether each function graphed or defined is one-to-one. $y=3 x^{3}-6$
Determine whether each function graphed or defined is one-to-one $y=\frac{-1}{x+2}$
Determine whether each function graphed or defined is one-to-one. $y=\frac{4}{x-8}$
Determine whether each function graphed or defined is one-to-one. $y=\frac{x+4}{x-3}$
Determine whether each function graphed or defined is one-to-one. $y=\frac{x-8}{x+1}$
Determine whether each function graphed or defined is one-to-one. $y=2(x+1)^{2}-6$
Determine whether each function graphed or defined is one-to-one $y=-3(x-6)^{2}+8$
Determine whether each function graphed or defined is one-to-one. $y=-\sqrt{x}+5$
Determine whether each function graphed or defined is one-to-one. and 2 y=\sqrt{x+3}-2
Determine whether each function graphed or defined is one-to-one. $y=5|x+2|$
Determine whether each function graphed or defined is one-to-one. $y=-|x|-4$
Determine whether each function graphed or defined is one-to-one. $y=\sqrt[3]{x+1}-3$
Determine whether each function graphed or defined is one-to-one. $y=-\sqrt[3]{x+2}-8$
Can a constant function, such as $f(x)=3,$ defined over the set of real numbers, be one-to-one?
Can a polynomial function of even degree defined over the set of real numbers have an inverse?
Use the definition of inverses to determine whether $f$ and $g$ are inverses $f(x)=2 x+4, \quad g(x)=\frac{1}{2} x-2$
Use the definition of inverses to determine whether $f$ and $g$ are inverses. $f(x)=3 x+9, \quad g(x)=\frac{1}{3} x-3$
Use the definition of inverses to determine whether $f$ and $g$ are inverses $f(x)=-3 x+12, \quad g(x)=-\frac{1}{3} x-12$
Use the definition of inverses to determine whether $f$ and $g$ are inverses $f(x)=-4 x+2, \quad g(x)=-\frac{1}{4} x-2$
Use the definition of inverses to determine whether $f$ and $g$ are inverses. $f(x)=\frac{x+1}{x-2}, \quad g(x)=\frac{2 x+1}{x-1}$
Use the definition of inverses to determine whether $f$ and g are inverses. $f(x)=\frac{x-3}{x+4}, \quad g(x)=\frac{4 x+3}{1-x}$
Use the definition of inverses to determine whether $f$ and $g$ are inverses $f(x)=\frac{2}{x+6}, \quad g(x)=\frac{6 x+2}{x}$
Use the definition of inverses to determine whether $f$ and $g$ are inverses. $f(x)=\frac{-1}{x+1}, \quad g(x)=\frac{1-x}{x}$
Use the definition of inverses to determine whether $f$ and $g$ are inverses. $f(x)=x^{2}+3, \quad x \geq 0 ; \quad g(x)=\sqrt{x-3}, \quad x \geq 3$
Use the definition of inverses to determine whether $f$ and $g$ are inverses. $f(x)=\sqrt{x+8}, \quad x \geq-8 ; \quad g(x)=x^{2}-8, \quad x \geq 0$
Determine whether the given functions are inverses $$\begin{array}{c|rrr|r}x & f(x) & & x & g(x) \\\hline 3 & -4 & & -4 & 3 \\2 & -6 & & -6 & 2 \\5 & 8 & & 8 & 5 \\1 & 9 & & 9 & 1 \\4 & 3 && 3 & 4\end{array}$$
Determine whether the given functions are inverses
Determine whether the given functions are inverses. $f=\{(2,5),(3,5),(4,5)\} ; \quad g=\{(5,2)\}$
Determine whether the given functions are inverses. $f=\{(1,1),(3,3),(5,5)\} ; \quad g=\{(1,1),(3,3),(5,5)\}$
Find the inverse of each function that is one-to-one. $$\{(-3,6),(2,1),(5,8)\}$$
Find the inverse of each function that is one-to-one. $\left\{(3,-1),(5,0),(0,5),\left(4, \frac{2}{3}\right)\right\}$
Find the inverse of each function that is one-to-one. $$\{(1,-3),(2,-7),(4,-3),(5,-5)\}$$
Find the inverse of each function that is one-to-one. $$\{(6,-8),(3,-4),(0,-8),(5,-4)\}$$
Determine whether each pair of functions graphed are inverses.
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and $(c)$ give the domain and range of both $f$ and $f^{-1}$. If the function is not one-to-one, say so. $f(x)=3 x-4$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and $(c)$ give the domain and range of both $f$ and $f^{-1}$. If the function is not one-to-one, say so. $f(x)=4 x-5$
For each function that is one-to-one, $(a)$ write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and $(c)$ give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=-4 x+3$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=-6 x-8$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=x^{3}+1$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=-x^{3}-2$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=x^{2}+8$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=-x^{2}+2$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=\frac{1}{x}, \quad x \neq 0$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=\frac{4}{x}, \quad x \neq 0$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=\frac{1}{x-3}, \quad x \neq 3$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=\frac{1}{x+2}, \quad x \neq-2$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=\frac{x+1}{x-3}, \quad x \neq 3$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=\frac{x+2}{x-1}, \quad x \neq 1$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=\frac{2 x+6}{x-3}, \quad x \neq 3$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=\frac{-3 x+12}{x-6}, \quad x \neq 6$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=\sqrt{x+6}, \quad x \geq-6$$
For each function that is one-to-one, (a) write an equation for the inverse function, (b) graph $f$ and $f^{-1}$ on the same axes, and (c) give the domain and range of both $f$ and $f^{-1} .$ If the function is not one-to-one, say so. $$f(x)=-\sqrt{x^{2}-16}, \quad x \geq 4$$
Graph the inverse of each one-to-one function.
The graph of a function $f$ is shown in the figure. Use the graph to find each value.$$f^{-1}(4)$$
The graph of a function $f$ is shown in the figure. Use the graph to find each value.$$f^{-1}(2)$$
The graph of a function $f$ is shown in the figure. Use the graph to find each value.$$f^{-1}(0)$$
The graph of a function $f$ is shown in the figure. Use the graph to find each value.$$f^{-1}(-2)$$
The graph of a function $f$ is shown in the figure. Use the graph to find each value.$$f^{-1}(-3)$$
The graph of a function $f$ is shown in the figure. Use the graph to find each value.$$f^{-1}(-4)$$
Suppose $f(x)$ is the number of cars that can be built for $x$ dollars. What does $f^{-1}(1000)$ represent?
Suppose $f(r)$ is the volume (in cubic inches) of a sphere of radius $r$ inches. What does $f^{-1}(5)$ represent?
Show that any function of the form $f(x)=-x+b$ is its own inverse.
For a one-to-one function $f,$ find $\left(f^{-1} \circ f\right)(2),$ where $f(2)=3$.
Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse.$$\begin{array}{l}f(x)=6 x^{3}+11 x^{2}-6 \\{[-3,2] \text { by }[-10,10]}\end{array}$$
Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse.$$\begin{array}{l}f(x)=x^{4}-5 x^{2} \\{[-3,3] \text { by }[-8,8]}\end{array}$$
Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse.$$\begin{array}{l}f(x)=\frac{x-5}{x+3}, \quad x \neq-3 \\{[-8,8] \text { by }[-6,8]}\end{array}$$
Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse.$$\begin{array}{l}f(x)=\frac{-x}{x-4}, \quad x \neq 4 \\{[-1,8] \text { by }[-6,6]}\end{array}$$
Use the following alphabet coding assignment to work each problem.$$\begin{array}{llllllll}\text { A } & 1 & \text { H } & 8 & \text { O } & 15 & \text { V } & 22 \\\text { B } & 2 & \text { I } & 9 & \text { P } & 16 & \text { W } & 23 \\\text { C } & 3 & \text { J } & 10 & \text { Q } & 17 & \text { X } & 24 \\\text { D } & 4 & \text { K } & 11 & \text { R } & 18 & \text { Y } & 25 \\\text { E } & 5 & \text { L } & 12 & \text { S } & 19 & \text { Z } & 26 \\\text { F } & 6 & \text { M } & 13 & \text { T } & 20 & & \\\text { G } & 7 & \text { N } & 14 & \text { U } & 21 & &\end{array}$$The function $f(x)=3 x-2$ was used to encode a message as$\begin{array}{lllllllllllll}37 & 25 & 19 & 61 & 13 & 34 & 22 & 1 & 55 & 1 & 52 & 52 & 25\end{array}$64Find the inverse function and determine the message.
Use the following alphabet coding assignment to work each problem.$$\begin{array}{llllllll}\text { A } & 1 & \text { H } & 8 & \text { O } & 15 & \text { V } & 22 \\\text { B } & 2 & \text { I } & 9 & \text { P } & 16 & \text { W } & 23 \\\text { C } & 3 & \text { J } & 10 & \text { Q } & 17 & \text { X } & 24 \\\text { D } & 4 & \text { K } & 11 & \text { R } & 18 & \text { Y } & 25 \\\text { E } & 5 & \text { L } & 12 & \text { S } & 19 & \text { Z } & 26 \\\text { F } & 6 & \text { M } & 13 & \text { T } & 20 & & \\\text { G } & 7 & \text { N } & 14 & \text { U } & 21 & &\end{array}$$The function $f(x)=2 x-9$ was used to encode a message as$$\begin{array}{llllllllllll}7 & -7 & 35 & 1 & -7 & 19 & 9 & -3 & 1 & -1 & -7 & 41\end{array}$$Find the inverse function and determine the message.
Use the following alphabet coding assignment to work each problem.$$\begin{array}{llllllll}\text { A } & 1 & \text { H } & 8 & \text { O } & 15 & \text { V } & 22 \\\text { B } & 2 & \text { I } & 9 & \text { P } & 16 & \text { W } & 23 \\\text { C } & 3 & \text { J } & 10 & \text { Q } & 17 & \text { X } & 24 \\\text { D } & 4 & \text { K } & 11 & \text { R } & 18 & \text { Y } & 25 \\\text { E } & 5 & \text { L } & 12 & \text { S } & 19 & \text { Z } & 26 \\\text { F } & 6 & \text { M } & 13 & \text { T } & 20 & & \\\text { G } & 7 & \text { N } & 14 & \text { U } & 21 & &\end{array}$$Encode the message SEND HELP, using the one-to-one function$$f(x)=x^{3}-1$$Give the inverse function that the decoder will need when the message is received.
Use the following alphabet coding assignment to work each problem.$$\begin{array}{llllllll}\text { A } & 1 & \text { H } & 8 & \text { O } & 15 & \text { V } & 22 \\\text { B } & 2 & \text { I } & 9 & \text { P } & 16 & \text { W } & 23 \\\text { C } & 3 & \text { J } & 10 & \text { Q } & 17 & \text { X } & 24 \\\text { D } & 4 & \text { K } & 11 & \text { R } & 18 & \text { Y } & 25 \\\text { E } & 5 & \text { L } & 12 & \text { S } & 19 & \text { Z } & 26 \\\text { F } & 6 & \text { M } & 13 & \text { T } & 20 & & \\\text { G } & 7 & \text { N } & 14 & \text { U } & 21 & &\end{array}$$Encode the message SAILOR BEWARE, using the one-to-one function$$f(x)=(x+1)^{3}$$Give the inverse function that the decoder will need when the message is received.