# 3.7 Inverse functions  (Page 7/9)

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For the following exercises, use function composition to verify that $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ are inverse functions.

$f\left(x\right)=\sqrt[3]{x-1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={x}^{3}+1$

$f\left(x\right)=-3x+5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\frac{x-5}{-3}$

## Graphical

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

$f\left(x\right)=\sqrt{x}$

one-to-one

$f\left(x\right)=\sqrt[3]{3x+1}$

$f\left(x\right)=-5x+1$

one-to-one

$f\left(x\right)={x}^{3}-27$

For the following exercises, determine whether the graph represents a one-to-one function.

not one-to-one

For the following exercises, use the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ shown in [link] .

Find $\text{\hspace{0.17em}}f\left(0\right).$

$3$

Solve $\text{\hspace{0.17em}}f\left(x\right)=0.$

Find $\text{\hspace{0.17em}}{f}^{-1}\left(0\right).$

$2$

Solve $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=0.$

For the following exercises, use the graph of the one-to-one function shown in [link] .

Sketch the graph of $\text{\hspace{0.17em}}{f}^{-1}.\text{\hspace{0.17em}}$

Find

If the complete graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is shown, find the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$

$\left[2,10\right]$

If the complete graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is shown, find the range of $\text{\hspace{0.17em}}f.$

## Numeric

For the following exercises, evaluate or solve, assuming that the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is one-to-one.

If $\text{\hspace{0.17em}}f\left(6\right)=7,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\text{\hspace{0.17em}}{f}^{-1}\left(7\right).$

$6$

If $\text{\hspace{0.17em}}f\left(3\right)=2,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}{f}^{-1}\left(2\right).$

If $\text{\hspace{0.17em}}{f}^{-1}\left(-4\right)=-8,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}f\left(-8\right).$

$-4$

If $\text{\hspace{0.17em}}{f}^{-1}\left(-2\right)=-1,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}f\left(-1\right).$

For the following exercises, use the values listed in [link] to evaluate or solve.

$x$ $f\left(x\right)$
0 8
1 0
2 7
3 4
4 2
5 6
6 5
7 3
8 9
9 1

Find $\text{\hspace{0.17em}}f\left(1\right).$

$0$

Solve $\text{\hspace{0.17em}}f\left(x\right)=3.$

Find $\text{\hspace{0.17em}}{f}^{-1}\left(0\right).$

$\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$

Solve $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=7.$

Use the tabular representation of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ in [link] to create a table for $\text{\hspace{0.17em}}{f}^{-1}\left(x\right).$

 $x$ 3 6 9 13 14 $f\left(x\right)$ 1 4 7 12 16
 $x$ 1 4 7 12 16 ${f}^{-1}\left(x\right)$ 3 6 9 13 14

## Technology

For the following exercises, find the inverse function. Then, graph the function and its inverse.

$f\left(x\right)=\frac{3}{x-2}$

$f\left(x\right)={x}^{3}-1$

${f}^{-1}\left(x\right)={\left(1+x\right)}^{1/3}$

Find the inverse function of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x-1}.\text{\hspace{0.17em}}$ Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

## Real-world applications

To convert from $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ degrees Celsius to $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ degrees Fahrenheit, we use the formula $\text{\hspace{0.17em}}f\left(x\right)=\frac{9}{5}x+32.\text{\hspace{0.17em}}$ Find the inverse function, if it exists, and explain its meaning.

${f}^{-1}\left(x\right)=\frac{5}{9}\left(x-32\right).\text{\hspace{0.17em}}$ Given the Fahrenheit temperature, $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ this formula allows you to calculate the Celsius temperature.

The circumference $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of a circle is a function of its radius given by $\text{\hspace{0.17em}}C\left(r\right)=2\pi r.\text{\hspace{0.17em}}$ Express the radius of a circle as a function of its circumference. Call this function $\text{\hspace{0.17em}}r\left(C\right).\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}r\left(36\pi \right)\text{\hspace{0.17em}}$ and interpret its meaning.

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ in hours given by $\text{\hspace{0.17em}}d\left(t\right)=50t.\text{\hspace{0.17em}}$ Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function $\text{\hspace{0.17em}}t\left(d\right).\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}t\left(180\right)\text{\hspace{0.17em}}$ and interpret its meaning.

$t\left(d\right)=\frac{d}{50},\text{\hspace{0.17em}}$ $t\left(180\right)=\frac{180}{50}.\text{\hspace{0.17em}}$ The time for the car to travel 180 miles is 3.6 hours.

## Functions and Function Notation

For the following exercises, determine whether the relation is a function.

$\left\{\left(a,b\right),\left(c,d\right),\left(e,d\right)\right\}$

function

$\left\{\left(5,2\right),\left(6,1\right),\left(6,2\right),\left(4,8\right)\right\}$

${y}^{2}+4=x,\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ the independent variable and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ the dependent variable

not a function

Is the graph in [link] a function?

For the following exercises, evaluate the function at the indicated values: $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(-3\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(2\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(-a\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-f\left(a\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(a+h\right).$

$f\left(x\right)=-2{x}^{2}+3x$

$f\left(-3\right)=-27;$ $f\left(2\right)=-2;$ $f\left(-a\right)=-2{a}^{2}-3a;$
$-f\left(a\right)=2{a}^{2}-3a;$ $f\left(a+h\right)=-2{a}^{2}+3a-4ah+3h-2{h}^{2}$

$f\left(x\right)=2|3x-1|$

For the following exercises, determine whether the functions are one-to-one.

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