5.1 Inverse Functions

Introduction

When you buy a house, your monthly mortgage payment is a function of the size of the loan. The table shows mortgage payments on $30$ -year loans of various sizes at $6$ % interest.

Loan amount, $L$	$150, 000$	$175, 000$	$200, 000$	$225, 000$	$250, 000$
Mortgage payment, $M$	$899.33$	$1049.21$	$1199.10$	$1348.99$	$1498.88$

For the function $M = f (L)$ , the input value is the amount of the loan, and the output is the mortgage payment.

However, when you are shopping for a house, you may think of the mortgage payment as the input variable: If you can afford a certain monthly mortgage payment, how large a loan can you finance? Now the mortgage payment is the input value, and the loan amount is the output. By interchanging the inputs and outputs, we define a new function, $L = g (M)$ , shown below.

Mortgage payment, $M$	$899.33$	$1049.21$	$1199.10$	$1348.99$	$1498.88$
Loan amount, $L$	$150, 000$	$175, 000$	$200, 000$	$225, 000$	$250, 000$

This new function gives the same information as the original function, $f$ , but from a different point of view. We call the function $g$ the inverse function for $f$ .

The elements of the range of $f$ are used as the input values for $g$ , and the output values of $g$ are the corresponding domain elements of $f$ . For example, from the tables you can verify that $f open parenthesis 200, 000 close parenthesis equals 1199.10$ , and $g open parenthesis 1199.10 close parenthesis equals 200, 000$ . In fact, this property defines the inverse function.

Example 1

Suppose $g$ is the inverse function for $f$ , and we know the following function values for $f$ :

$f (- 3) = 5, f (2) = 1, f (5) = 0$

Find $g open parenthesis 5 close parenthesis$ and $g open parenthesis 0 close parenthesis$ .

We know that $g (5) = - 3$ because $f (- 3) = 5$ , and $g open parenthesis 0 close parenthesis equals 5$ because $f open parenthesis 5 close parenthesis equals 0$ . Tables may be helpful in visualizing the two functions, as shown below.

y = f (x)

x

y

minus 3

5

2

1

5

0

$→ Interchange the columns →$

x = g (y)

y

x

5

minus 3

1

2

0

5

For the function $f$ , the input variable is $x$ and the output variable is $y$ . For the inverse function $g$ , the roles of the variables are interchanged: $y$ is now the input and $x$ is the output.

Suppose $g$ is the inverse function for $f$ , and suppose we know the following function values for f:

$f (- 1) = 0, f (0) = 1, f (1) = 2$

Find $g open parenthesis 0 close parenthesis$ and $g open parenthesis 1 close parenthesis$ .

$g open parenthesis 0 close parenthesis equals$ _____

$g open parenthesis 1 close parenthesis equals$ _____

$g (0) = - 1$ ; $g open parenthesis 1 close parenthesis equals 0$

Suppose $g$ is the inverse function for $f$ , and suppose we know the following function values for $f$ :

$f (- 1) = 0, f (0) = 1, f (1) = 2$

Find $g open parenthesis 0 close parenthesis$ and $g open parenthesis 1 close parenthesis$ .

$g (0) = - 1$ ; $g open parenthesis 1 close parenthesis equals 0$

If the point $open parenthesis 3, minus 4 close parenthesis$ lies on the graph of $f$ , what point lies on the graph of its inverse function?

_____

$open parenthesis minus 4, 3 close parenthesis$

If the point $open parenthesis 3, minus 4 close parenthesis$ lies on the graph of $f$ , what point lies on the graph of its inverse function?

$open parenthesis minus 3, 4 close parenthesis$
$open parenthesis minus 4, 3 close parenthesis$
$open parenthesis 4, minus 3 close parenthesis$
$(\frac{1}{3}, - \frac{1}{4})$

Finding a Formula for the Inverse Function

If a function is given by a table of values, we can interchange the columns (or rows) of the table to obtain the inverse function. Swapping the columns works because we are really interchanging the input and output variables. If a function is defined by an equation, we can find a formula for its inverse function in the same way: Interchange the roles of the variables in the equation so that the old output variable becomes the new input variable.

Example 2

The function $H = f (t) = 6 + 2 t$ gives the height of corn seedlings, in inches, $t$ days after they are planted. Find a formula for the inverse function and explain its meaning in this context.
Make a table of values for $f (t)$ and a table for its inverse function.

Write the equation for $f$ in the form
$H = 6 + 2 t$
In this equation, $t$ is the input and $H$ is the output. We interchange the roles of the variables by solving for $t$ to obtain
$t = \frac{H - 6}{2}$
In this equation, $H$ is the input and $t$ is the output. The formula for the inverse function is
$t = g (H) = \frac{H - 6}{2}$
The function $g$ gives the number of days it will take the corn seedlings to grow to a height of $H$ inches.
To make a table for $f$ , we choose values for $t$ and evaluate $f (t) = 6 + 2 t$ at those $t$ -values, as shown at left below.
$H = f (t)$
$t$ $H$
$0$ $6$
$1$ $8$
$2$ $10$
$3$ $12$
$t = g (H)$
$H$ $t$
$6$ $0$
$8$ $1$
$10$ $2$
$12$ $3$

To make a table for $g$ , we could choose values for $H$ and evaluate $\frac{H - 6}{2}$ , but because $g$ is the inverse function for $f$ , we can simply interchange the columns in our table for $f$ , as shown at right above.
You can check that the values in the second table do satisfy the formula for the inverse function, $g (H) = \frac{H - 6}{2}$ .

Which of these is the inverse of the function $h = 3 m - 2$ ?

_____

$m = \frac{h + 2}{3}$

Which of these is the inverse of the function $h = 3 m - 2$ ?

$m = \frac{h + 2}{3}$
$m = 2 h - 3$
$m = \frac{h - 3}{2}$
$m = \frac{1}{3 h - 2}$

Carol can burn 600 calories per hour bicycling and 400 calories per hour swimming. She would like to lose 5 pounds, which is equivalent to 16,000 calories.

Write an equation relating the number of hours of cycling, $x$ , and the number of hours swimming, $y$ , that Carol must spend to lose 5 pounds. [Do not use commas: For example, enter "10000" rather than "10,000".
Answer: _____
Write $y$ as a function of $x$ , $y = f (x)$ .
$y = f (x) =$ _____
What does $f open parenthesis 10 close parenthesis equals$ _____ tell you?
_____
Find the inverse function, $x = g (y)$ .
$x = g (y) =$ _____
What does $g open parenthesis 10 close parenthesis equals$ _____ tell you?
_____

$600 x + 400 y = 16, 000$
$y = f (x) = 40 - 1.5 x$ ; $f open parenthesis 10 close parenthesis equals 25$ ; If Carol cycles for 10 hrs, she must swim for 25 hrs.
$x = g (y) = 26 \frac{2}{3} - \frac{2}{3} y$ ; $g open parenthesis 10 close parenthesis equals 20$ ; If Carol swims for $10$ hrs, she must cycle for $20$ hrs.

Carol can burn 600 calories per hour bicycling and 400 calories per hour swimming. She would like to lose 5 pounds, which is equivalent to 16,000 calories.

Write an equation relating the number of hours of cycling, $x$ , and the number of hours swimming, $y$ , that Carol must spend to lose 5 pounds.
Write $y$ as a function of $x$ , $y = f (x)$ . What does $f open parenthesis 10 close parenthesis$ tell you?
Find the inverse function, $x = g (y)$ . What does $g open parenthesis 10 close parenthesis$ tell you?

$600 x + 400 y = 16, 000$
$y = f (x) = 40 - 1.5 x$ ; $f open parenthesis 10 close parenthesis equals 25$ ; If Carol cycles for 10 hrs, she must swim for 25 hrs.
$x = g (y) = 26 \frac{2}{3} - \frac{2}{3} y$ ; $g open parenthesis 10 close parenthesis equals 20$ ; If Carol swims for $10$ hrs, she must cycle for $20$ hrs.

Inverse Function Notation

If the inverse of a function $f$ is also a function, we denote the inverse by the symbol $f^{- 1}$ , read " $f$ inverse." This notation makes it clear that the two functions are related in a special way. For example, the function $f (t) = 6 + 2 t$ in Example has inverse function $f^{- 1} (H) = \frac{H - 6}{2}$ .

Example 3

If $y = f (x) = x^{3} + 2,$ find $f^{- 1} (10)$ .

We first find the inverse function for $y = x^{3} + 2$ by solving for $x$ :

$\begin{aligned} x^{3} & = y - 2 & Substract 2 from both sides. \\ x & = \sqrt[3]{y - 2} & Take cube roots. \end{aligned}$

The inverse function is $x = f^{- 1} (y) = \sqrt[3]{y - 2}$ . Now we evaluate the inverse function at $y equals 10$ :

$f^{- 1} (10) = \sqrt[3]{10 - 2} = 2$

Caution

Although the same symbol, $^{- 1}$ , is used for both reciprocals and inverse functions, the two notions are not equivalent. That is, the inverse of a given function is usually not the same as the reciprocal of that function. In Example, note that $f^{- 1} (y)$ is not the same as the reciprocal of $f (y)$ , because

$\frac{1}{f (y)} = \frac{1}{y^{3} + 2} but f^{- 1} (y) = \sqrt[3]{y - 2}$

To avoid confusion, we use the notation $\frac{1}{f}$ to refer to the reciprocal of the function $f$ .

If $f open parenthesis 5 close parenthesis equals 2$ , what is $f^{- 1} (2)$ equal to?

_____

$5$

If $f open parenthesis 5 close parenthesis equals 2$ , what is $f^{- 1} (2)$ equal to?

$\frac{1}{2}$
$\frac{1}{5}$
$5$
$minus 5$

If $z = f (w) = \frac{1}{w + 3}$ , find $f^{- 1} (1)$ .
$f^{- 1} (1) =$ _____
Write two equations about the value of $f^{- 1} (1)$ , one using $f^{- 1}$ and one using $f$ .
$f^{- 1} (1) =$ _____, and $f (- 2) =$ _____
Show that $f^{- 1} (1)$ is not equal to $\frac{1}{f (1)}$ .
$f^{- 1} (1) =$ _____, but $\frac{1}{f (1)} =$ _____, which _____

$minus 2$
$f^{- 1} (1) = - 2$ , $f (- 2) = 1$
$f^{- 1} (1) = - 2$ , but $\frac{1}{f (1)} = 4$

If $z = f (w) = \frac{1}{w + 3}$ , find $f^{- 1} (1)$ .
Write two equations about the value of $f^{- 1} (1)$ , one using $f^{- 1}$ and one using $f$ .
Show that $f^{- 1} (1)$ is not equal to $\frac{1}{f (1)}$ .

$minus 2$
$f^{- 1} (1) = - 2$ , $f (- 2) = 1$
$f^{- 1} (1) = - 2$ , but $\frac{1}{f (1)} = 4$

We can use a graph of a function $y = f (x)$ to find values of the inverse function $x = f^{- 1} (y)$ . The figure below shows the graph of $f (x) = x^{3} + 2$ .

You already know how to evaluate a function from its graph: We start with the horizontal axis. For instance, to evaluate $f open parenthesis minus 2 close parenthesis$ , we find $minus 2$ on the $x$ -axis, move vertically to the point on the graph with $x = - 2$ , in this case $(- 2, - 6)$ , and read the $y$ -coordinate of the point. We see that $f (- 2) = - 6$ .

To evaluate the inverse function, we start with the vertical axis. For example, to find $f^{- 1} (10)$ , we find $10$ on the vertical axis and move horizontally to the point on the graph with $y equals 10$ . In this case, the point is $open parenthesis 2, 10 close parenthesis$ , so $f^{- 1} (10) = 2$ .

reading values of f-inverse from graph of f

Example 4

The function $C = h (F)$ gives Celsius temperature as a function of Fahrenheit temperature. The graph of the function is shown below. Use the graph to evaluate $h open parenthesis 68 close parenthesis$ and $h^{- 1} (10)$ , and then explain their meaning in this context.

To evaluate $h open parenthesis 68 close parenthesis$ , we find the input $F equals 68$ on the horizontal axis, then find the point on the graph with $F equals 68$ and read its vertical coordinate. We see that the point $open parenthesis 68, 20 close parenthesis$ lies on the graph, so $h open parenthesis 68 close parenthesis equals 20$ . When the Fahrenheit temperature is $68 degrees$ , the Celsius temperature is $20 degrees$ .

The inverse function reverses the roles of input and output. Because $C = h (F)$ , $F = h^{- 1} (C)$ , so the inverse function gives us the Fahrenheit temperature if we know the Celsius temperature. In particular, $h^{- 1} (10)$ is the Fahrenheit temperature when the Celsius temperature is $10 degrees$ .

To use the graph of $h$ to find values of $h^{- 1}$ , we start with the vertical axis and find the point on the graph with $C equals 10$ . This point is $open parenthesis 50, 10 close parenthesis$ , so $F equals 50$ when $C equals 10$ , or $h^{- 1} (10) = 50$ . When the Celsius temperature is $10 degrees$ , the Fahrenheit temperature is $50 degrees$ .

Use the graph of $h$ in Example.

Find $h^{- 1} (- 10)$ .
$h^{- 1} (- 10) =$ _____
Does $h^{- 1} (- 10) = - h^{- 1} (10)$ ? _____
Select two equations, one using h and one using h − 1 , stating the Fahrenheit temperature when the Celsius temperature is 0 ∘ .
- $h open parenthesis 32 close parenthesis equals 0$ _____
- $h open parenthesis 0 close parenthesis equals 32$ _____
- $h^{- 1} (32) = 0$ _____
- $h^{- 1} (0) = 32$ _____

$14$ : On the graph of $h$ , when $C = - 10$ , $F equals 14$ .
No
$h open parenthesis 32 close parenthesis equals 0$ , $h^{- 1} (0) = 32$

Use the graph of $h$ in Example to find $h^{- 1} (- 10)$ .
Does $h^{- 1} (- 10) = - h^{- 1} (10)$ ?
Write two equations, one using $h$ and one using $h^{- 1}$ , stating the Fahrenheit temperature when the Celsius temperature is $0 degrees$ .

$14$ : On the graph of $h$ , when $C = - 10$ , $F equals 14$ .
No
$h open parenthesis 32 close parenthesis equals 0$ , $h^{- 1} (0) = 32$

If the function $f$ tells you how much money is in your account after $t$ years, what does the function $f^{- 1}$ tell you?

_____

If the function $f$ tells you how much money is in your account after $t$ years, what does the function $f^{- 1}$ tell you?

Graph of the Inverse Function

In Example, we used a graph of $h$ to read values of $h^{- 1}$ . But we can also plot the graph of $h^{- 1}$ itself. Because $C$ is the input variable for $h^{- 1}$ , we plot $C$ on the horizontal axis and $F$ on the vertical axis. To find some points on the graph of $h^{- 1}$ , we interchange the coordinates of points on the graph of $h$ . The graph of $h^{- 1}$ is shown at right.

C = h (F)

F = h^{- 1} (C)

F

C

C

F

14

minus 10

minus 10

14

32

0

0

32

50

10

10

50

68

20

20

68

Example 5

The Park Service introduced a flock of $12$ endangered pheasant into a wildlife preserve. After $t$ years, the population of the flock was given by

$P = f (t) = 12 + 2 t^{3}$

Graph the function on the domain $[ 0, 5 ]$ .
Find a formula for the inverse function, $t = f^{- 1} (P)$ . What is the meaning of the inverse function in this context?
Sketch a graph of the inverse function.

The graph of $f$ is shown at right, with $t$ on the horizontal axis and $P$ on the vertical axis.
We solve $P = 12 + 2 t^{3}$ for $t$ in terms of $P$ .
$\begin{aligned} 2 t^{3} & = P - 12 & Substract 12 from both sides. \\ t^{3} & = \frac{P - 12}{2} & Divide both sides by 2. \\ t & = \sqrt[3]{\frac{P - 12}{2}} & Take cube roots. \end{aligned}$
The inverse function is $t = f^{- 1} (P) = \sqrt[3]{\frac{P - 12}{2}}$ . It tells us the number of years it takes for the pheasant population to grow to size $P$ .
The graph of $f^{- 1}$ is shown below, with $P$ on the horizontal axis and $t$ on the vertical axis.

The formula $T = f (L) = 2 π \sqrt{\frac{L}{32}}$ gives the period in seconds, $T$ , of a pendulum as a function of its length in feet, $L$ .

Graph the function on the domain $[ 0, 5 ]$ .
Find a formula for the inverse function, $L = f^{- 1} (T)$ . [Enter "pi" to get $π$ .]
$L = f^{- 1} (T) =$ _____
What is the meaning of the inverse function in this context?
_____
Sketch a graph of the inverse function.

A graph is below for both parts (a) and (c).
$L = f^{- 1} (T) = \frac{8}{π^{2}} T^{2} .$ $f^{- 1}$ gives the length of a pendulum as a function of its period.
See graph for part(a) below.

The formula $T = f (L) = 2 π \sqrt{\frac{L}{32}}$ gives the period in seconds, $T$ , of a pendulum as a function of its length in feet, $L$ .

Graph the function on the domain $[ 0, 5 ]$ .
Find a formula for the inverse function, $L = f^{- 1} (T)$ . What is the meaning of the inverse function in this context?
Sketch a graph of the inverse function.

$L = f^{- 1} (T) = \frac{8}{π^{2}} T^{2} .$ $f^{- 1}$ gives the length of a pendulum as a function of its period.
See graph for part(a).

When Is the Inverse a Function?

We can always find the inverse of a function simply by interchanging the input and output variables. In the preceding examples, interchanging the variables created a new function. However, the inverse of a function does not always turn out to be a function itself.

For example, to find the inverse of $y = f (x) = x^{2}$ , we solve for $x$ to get $x = \pm \sqrt{y}$ . When we regard $y$ as the input and $x$ as the output, the relationship does not describe a function. The graphs of $f$ and its inverse are shown below. (Note that for the graph of the inverse, we plot $y$ on the horizontal axis and $x$ on the vertical axis.) Because the graph of the inverse does not pass the vertical line test, it is not a function.

For many applications, it is important to know whether or not the inverse of $f$ is a function. This can be determined from the graph of $f$ . When we interchange the roles of the input and output variables, horizontal lines of the form $y = k$ become vertical lines.

Thus, if the graph of the inverse is going to pass the vertical line test, the graph of the original function must pass the horizontal line test, namely, that no horizontal line should intersect the graph in more than one point.

Notice that the graph of $f (x) = x^{2}$ does not pass the horizontal line test, so we would not expect its inverse to be a function.

Which of the functions whose graphs are shown above have inverses that are also functions?

_____
_____
_____
_____

(a) and (d)

Which of the functions whose graphs are shown below have inverses that are also functions?

(a) and (d)

A function that passes the horizontal line test is called one-to-one, because each input has only one output and each output has only one input. A one-to-one function passes the horizontal line test as well as the vertical line test. With this terminology, we can state the following theorem.

What does the term one-to-one mean? Give an example.

_____

What does the term one-to-one mean? Give an example.

Mathematical Properties of the Inverse Function

The inverse function $f^{- 1}$ undoes the effect of the function $f$ . In Example, the function $f (t) = 6 + 2 t$ multiplies the input by $2$ and then adds $6$ to the result. The inverse function $f^{- 1} (H) = \frac{H - 6}{2}$ undoes those operations in reverse order: It subtracts $6$ from the input and then divides the result by $2$ .

If we apply the function $f$ to a given input value and then apply the function $f^{- 1}$ to the output from $f$ , the end result will be the original input value. For example, if we choose $t equals 5$ as an input value, we find that

$\begin{aligned} f (5) & = 6 + 2 (5) = 16 & Multiply by 2, then add 6. \\ and f^{- 1} (16) & = \frac{16 - 6}{2} = 5 & Subtract 6, then divide by 2. \end{aligned}$

We return to the original input value, $5$ , as illustrated above.

An inverse function

_____

interchanges the input and output.

An inverse function

makes the output negative.
makes the input negative.
interchanges the input and output.
takes the reciprocal of the output.

Example illustrates the fact that if $f^{- 1}$ is the inverse function for $f$ , then $f$ is also the inverse function for $f^{- 1}$ .

Example 7

Consider the function $f (x) = x^{3} + 2$ and its inverse, $f^{- 1} (y) = \sqrt[3]{y - 2}$ .

Show that the inverse function undoes the effect of $f$ on $x equals 2$ .
Show that $f$ undoes the effect of the inverse function on $y = - 25$ .

We first evaluate the function $f$ for $x equals 2$ :
$f (2) = 2^{3} + 2 = 10$
Then we evaluate the inverse function $f^{- 1}$ at $y equals 10$ :
$f^{- 1} (10) = \sqrt[3]{10 - 2} = \sqrt[3]{8} = 2$
We started and ended with $2$ .
We first evaluate the function $f^{- 1}$ for $y = - 25$ :
$f^{- 1} (- 25) = \sqrt[3]{- 25 - 2} = - 3$
Then we evaluate the function $f$ for $x = - 3$ :
$f (- 3) = (- 3)^{3} + 2 = - 25$
We started and ended with $minus 25$ .

Find a formula for the inverse of the function $f (x) = \frac{2}{x - 1}$ Use $y$ as in input variable for $f^{- 1}$ .
$f^{- 1} (y) =$ _____
Show that $f^{- 1}$ undoes the effect of $f$ on $x equals x1$ .
$f open parenthesis x1 close parenthesis equals$ _____ and $f^{- 1} (y1) =$ _____
Show that $f$ undoes the effect of $f^{- 1}$ on $y = - 2$ .
$f^{- 1} (y2) =$ _____ and $f open parenthesis x2 close parenthesis equals$ _____

$f^{- 1} (y) = 1 + \frac{2}{y}$
$f open parenthesis 3 close parenthesis equals 1$ , and $f^{- 1} (1) = 3$
$f^{- 1} (- 2) = 0$ and $f (0) = - 2$

Find a formula for the inverse of the function $f (x) = \frac{2}{x - 1}$ .
Show that $f^{- 1}$ undoes the effect of $f$ on $x equals 3$ .
Show that $f$ undoes the effect of $f^{- 1}$ on $y = - 2$ .

$f^{- 1} (y) = 1 + \frac{2}{y}$
$f open parenthesis 3 close parenthesis equals 1$ , and $f^{- 1} (1) = 3$
$f^{- 1} (- 2) = 0$ and $f (0) = - 2$

Symmetry

So far we have been careful to keep track of the input and output variables when we work with inverse functions. This is important when we are dealing with applications; the names of the variables are usually chosen because they have a meaning in the context of the application, and it would be confusing to change them.

However, we can also study inverse functions purely as mathematical objects. There is a relationship between the graph of a function and the graph of its inverse that is easier to see if we plot them both on the same set of axes.

A graph does not change if we change the names of the variables, so we can let $x$ represent the input for both functions, and let $y$ represent the output. Consider the function $C = h (F)$ from Example, and its inverse function, $F = h^{- 1} (C)$ . The formulas for these functions are

$\begin{aligned} C & = h (F) = \frac{5}{9} (F - 32) \\ F & = h^{- 1} (C) = 32 + \frac{9}{5} C \end{aligned}$

But their graphs are the same if we write them as

$\begin{aligned} y & = h (x) = \frac{5}{9} (x - 32) \\ y & = h^{- 1} (x) = 32 + \frac{9}{5} x \end{aligned}$

The graphs are shown below.

Now, for every point $(a, b)$ on the graph of $f$ , the point $(b, a)$ is on the graph of the inverse function. Observe that the points $(a, b)$ and $(b, a)$ are always located symmetrically across the line $y = x$ . The graphs are symmetric about the line $y = x$ , which means that if we were to place a mirror along the line $y = x$ , each graph would be the reflection of the other.

Function graph showing y = x, y = x^3 + a and the parametric curve (t^3 + a, t) for t in [-2.5, 2.5]. Adjustable parameter: Shift a (a) = 1. Viewing window: x from -4.87 to 4.87, y from -3.01 to 3.01. — The symmetry this paragraph proves, kept live under a slider. The blue curve is f(x) = x³ + a; the red curve is its inverse, drawn by interchanging the coordinates of every point on f — which is all an inverse is. The two curves are mirror images across the dashed line y = x for every value of a. Watch what the slider does to each: raising a shifts f upward, and shifts the inverse by the same amount to the right — because swapping coordinates turns a vertical shift into a horizontal one, an input change into an output change.

The graphs of $f$ and $f^{- 1}$ are symmetric about

_____

the line $y = x$ .

The graphs of $f$ and $f^{- 1}$ are symmetric about

the $x$ -axis
the $y$ -axis
the asymptote.
the line $y = x$ .

Example 8

Graph the function $f (x) = 2 \sqrt{x + 4}$ on the domain $[ minus 4, 12 ]$ . Graph its inverse function $f^{- 1}$ on the same grid.

The graph of $f$ has the same shape as the graph of $y = \sqrt{x}$ , shifted $4$ units to the left and stretched vertically by a factor of $2$ . The figure at left below shows the graph of $f$ , along with a table of values. By interchanging the rows of the table, we obtain points on the graph of the inverse function, shown at right.

If we use $x$ as the input variable for both functions, and $y$ as the output, we can graph $f$ and $f^{- 1}$ on the same grid, as shown at right. The two graphs are symmetric about the line $y = x$ .

Graph the function $f (x) = x^{3} + 2$ and its inverse $f^{- 1} (x) = \sqrt[3]{x - 2}$ on the same set of axes, along with the line $y = x$ .

Graph the function $f (x) = x^{3} + 2$ and its inverse $f^{- 1} (x) = \sqrt[3]{x - 2}$ on the same set of axes, along with the line $y = x$ .

Explain the difference between the meaning of the notation $p^{- 1}$ if $p$ is a function or if $p$ is a variable.

_____

Explain the difference between the meaning of the notation $p^{- 1}$ if $p$ is a function or if $p$ is a variable.

Domain and Range

When we interchange the input and output variables to obtain the inverse function, we interchange the domain and range of the function. For the functions graphed in Example, you can see that

$\begin{aligned} Domain (f) & = [- 4, 12] & and & Domain (f^{- 1}) & = [0, 8] \\ Range (f) & = [0, 8] & Range (f - 1) & = [- 4, 12] \end{aligned}$

This relationship between the domain and range of a function and its inverse holds in general.

Note

Domain and Range of the Inverse Function

If $f^{- 1}$ is the inverse function for $f,$ then

$\begin{aligned} Domain (f^{- 1}) & = Range (f) \\ Range (f^{- 1}) & = Domain (f) \end{aligned}$

Example 9

Graph the function $y = f (x) = \frac{1}{x + 3}$ in the window
$\begin{aligned} Xmin & = - 6 & Xmax = 3.4 \\ Ymin & = - 6 & Ymax = 3 \end{aligned}$
Graph the inverse function in the same window, along with the line $y = x$ .
State the domain and range of $f$ , and of $f^{- 1}$ .

The graph of $f$ is shown below. It looks like the graph of $y = \frac{1}{x}$ , shifted $3$ units to the left.
To find the inverse function, we solve for $x$ . Take the reciprocal of both sides of the equation.
$\begin{aligned} \frac{1}{y} & = x + 3 & Subtract 3 from both sides. \\ x & = \frac{1}{y} - 3 \end{aligned}$
The inverse function is $x = f^{- 1} (y) = \frac{1}{y} - 3$ , or, using $x$ for the input variable, $f^{- 1} (x) = \frac{1}{x} - 3$ . The graph of $f^{- 1}$ looks like the graph of $y = \frac{1}{x}$ , shifted down $3$ units, as shown below.
Because $f$ is undefined at $x = - 3$ , the domain of $f$ is all real numbers except $minus 3$ . The graph has a horizontal asymptote at $y equals 0$ , so the range is all real numbers except $0$ .
The inverse function $f^{- 1} (x) = \frac{1}{x} - 3$ is undefined at $x equals 0$ , so its domain is all real numbers except $0$ . The graph of $f^{- 1}$ has a horizontal asymptote at $y = - 3$ , so its range is all real numbers except $minus 3$ .

Graph the function $f (x) = \frac{2}{x - 1}$ and its inverse function, $f^{- 1}$ (which you found above in Practice 7, on the same set of axes, along with the line $y = x$ .
State the domain and range of $f$ , and of $f^{- 1}$ .
The domain of $f$ includes all real numbers except _____, and the range of $f$ includes all real number except _____
The domain of $f^{- 1}$ includes all real numbers except _____, and the range of $f^{- 1}$ includes all real number except _____

A graph is below.
Domain of $f$ : all real numbers except $1$ , Range of $f$ : all real numbers except $0$ , Domain of $f^{- 1}$ : all real numbers except $0$ , Range of $f^{- 1}$ : all real numbers except $1$

Another graph of $f$ and $f^{- 1}$

Graph the function $f (x) = \frac{2}{x - 1}$ and its inverse function, $f^{- 1}$ (which you found above in Practice 7, on the same set of axes, along with the line $y = x$ .
State the domain and range of $f$ , and of $f^{- 1}$ .

Domain of $f$ : all real numbers except $1$ , Range of $f$ : all real numbers except $0$ , Domain of $f^{- 1}$ : all real numbers except $0$ , Range of $f^{- 1}$ : all real numbers except $1$

Section Summary

Vocabulary

Look up the definitions of new terms in the Glossary.

Inverse function
Horizontal line test
One-to-one

CONCEPTS

The inverse of a function describes the same relationship between two variables but interchanges the roles of the input and output.
Note
Inverse Functions
If the inverse of a function $f$ is also a function, then the inverse is denoted by the symbol $f^{- 1}$ , and
$f^{- 1} (b) = a if and only if f (a) = b$
We can make a table of values for the inverse function, $f^{- 1}$ , by interchanging the columns of a table for $f$ .
If a function is defined by a formula in the form $y = f (x)$ , we can find a formula for its inverse function by solving the equation for $x$ to get $x = f^{- 1} (y)$ .
The inverse function $f^{- 1}$ undoes the effect of the function $f$ , that is, if we apply the inverse function to the output of $f$ , we return to the original input value.
If $f^{- 1}$ is the inverse function for $f$ , then $f$ is also the inverse function for $f^{- 1}$ .
The graphs of $f$ and its inverse function are symmetric about the line $y = x$ .
Horizontal line test: If no horizontal line intersects the graph of a function more than once, then the inverse is also a function.
A function that passes the horizontal line test is called one-to-one.
The inverse of a function $f$ is also a function if and only if $f$ is one-to-one.
Note
Functions and Inverse Functions
Suppose $f^{- 1}$ is the inverse function for $f$ . Then
$f^{- 1} (f (x)) = x and f (f^{- 1} (y)) = y$
as long as $x$ is in the domain of $f$ , and $y$ is in the domain of $f^{- 1}$ .
Note
Domain and Range of the Inverse Function
If $f^{- 1}$ is the inverse function for $f,$ then
$\begin{aligned} Domain (f^{- 1}) & = Range (f) \\ Range (f^{- 1}) & = Domain (f) \end{aligned}$

STUDY QUESTIONS

Explain how the terms inverse function, one-to-one, and horizontal line test are related.
If you know that $f^{- 1} (3) = - 7$ , what can you say about the values of $f$ ?
Explain how to use a graph of the function $g$ to evaluate $g^{- 1} (2)$ .
Evaluate $f (f^{- 1} (5))$ .
Delbert says that if $f (x) = x^{3 / 5}$ , then $f^{- 1} (x) = x^{- 3 / 5}$ . Is he correct? Why or why not?

SKILLS

Practice each skill in the Homework problems listed.

Given certain function values, find values of the inverse function: #1–4
Interpret values of the inverse function: #5–12
Find a formula for the inverse function: #9–22, 27–34
Graph the inverse function: #15 and 16, 23–34
Find the domain and range of the inverse function: #33 and 34
Use the horizontal line test to identify one-to-one functions: #35–42

Homework 5.1

Let $f (- 1) = 0$ , $f open parenthesis 0 close parenthesis equals 1$ , $f (1) = - 2$ , and $f (2) = - 1$ .

Make a table of values for $f (x)$ and another table for its inverse function.
Find $f^{- 1} (1)$
Find $f^{- 1} (- 1)$

$x$ $minus 1$ $0$ $1$ $2$
$f (x)$ $0$ $1$ $minus 2$ $minus 1$
$y$ $0$ $1$ $minus 2$ $minus 1$
$f^{- 1} (y)$ $minus 1$ $0$ $1$ $2$
$f^{- 1} (1) = 0$
$f^{- 1} (- 1) = 2$

$x$	$minus 1$	$0$	$1$	$2$
$f (x)$	$0$	$1$	$minus 2$	$minus 1$

$y$	$0$	$1$	$minus 2$	$minus 1$
$f^{- 1} (y)$	$minus 1$	$0$	$1$	$2$

Let $f (- 2) = 1$ , $f (- 1) = - 2$ , $f open parenthesis 0 close parenthesis equals 0$ , and $f (1) = - 1$ .

Make a table of values for $f (x)$ and another table for its inverse function.
Find $f^{- 1} (- 1)$
Find $f^{- 1} (1)$

$f (x) = x^{3} + x + 1$

Make a table of values for $f (x)$ and another table for its inverse function.
Find $f^{- 1} (1)$
Find $f^{- 1} (3)$

$x$ $minus 1$ $0$ $1$ $2$
$f (x)$ $minus 1$ $1$ $3$ $11$
$y$ $minus 1$ $1$ $3$ $11$
$f^{- 1} (y)$ $minus 1$ $0$ $1$ $2$
$f^{1} (1) = 0$
$f^{- 1} (3) = 1$

$x$	$minus 1$	$0$	$1$	$2$
$f (x)$	$minus 1$	$1$	$3$	$11$

$y$	$minus 1$	$1$	$3$	$11$
$f^{- 1} (y)$	$minus 1$	$0$	$1$	$2$

$f (x) = x^{5} + x^{3} + 7$

Make a table of values for $f (x)$ and another table for its inverse function.
Find $f^{- 1} (7)$
Find $f^{- 1} (5)$

For Problems 5-8, use the graph to evaluate each expression.

An insurance investigator measures the length, $d$ , of the skid marks at an accident scene, in feet. The graph shows the function $v = f (d)$ , which gives the velocity, $v$ (mph), at which a car was traveling when it hit the brakes.

Use the graph to estimate $f open parenthesis 60 close parenthesis$ and explain its meaning in this context.
Use the graph to estimate $f^{- 1} (60)$ and explain its meaning in this context.

$f open parenthesis 60 close parenthesis approximately equals 38$ . The car that left the $60$ -foot skid marks was traveling at $38$ mph.
$f^{- 1} (60) \approx 150$ . The car traveling at $60$ mph left $150$ -foot skid marks

The weight, $m$ , of a missile launched from a catapult is a function of the distance, $d$ , to the target. The graph shows the function $m = f (d)$ , where $d$ is in meters and $m$ is in kilograms.

Use the graph to estimate $f open parenthesis 100 close parenthesis$ and explain its meaning in this context.
Use the graph to estimate $f^{- 1} (100)$ and explain its meaning in this context.

After eating, the weight of a vampire bat drops steadily until its next meal. The graph shows the function $W = f (t)$ , which gives the weight, $W$ , of the bat in grams $t$ hours since its last meal.

Estimate the coordinates of the point of starvation. Include units in your answer.
Use the graph to estimate $f^{- 1} (90)$ and explain what it tells us about vampire bats.

$open parenthesis 60 hours, 78 grams close parenthesis$
$f^{- 1} (90) \approx 19$ , so that the vampire bat's weight has dropped to $90$ grams about $19$ hours after its last meal.

The amount of money, $A$ , in an interest-bearing savings account is a function of the number of years, $t$ , it remains in the account. The graph shows $A = f (t)$ , where $A$ is in thousands of dollars.

Use the graph to estimate $f open parenthesis 30 close parenthesis$ and explain what it tells us about the account.
Use the graph to estimate $f^{- 1} (30)$ and explain what it tells us about the account.

The function $I = g (r) = (1 + r)^{5} - 1$ gives the interest, $I$ , that a dollar earns in $5$ years in terms of the interest rate, $r$ .

Evaluate $g open parenthesis 0.05 close parenthesis$ and explain what it tells us about the interest.
Find the interest rate needed to earn $$ 0.50$ by substituting $I equals 0.50$ in the formula and solving for $r$ .
Find a formula for the inverse function.
Write your answer to part (b) with inverse function notation.

$g open parenthesis 0.05 close parenthesis equals 0.28$ . At $5 %$ interest, $$ 1$ earns $$ 0.28$ interest in $5$ years.
$8.45 %$
$g^{- 1} (I) = (I + 1)^{1 / 5} - 1$
$g^{- 1} (0.50) \approx 0.0845$

The function $C = h (F) = \frac{5}{9} (F - 32)$ gives the Celsius temperature $C$ in terms of the Fahrenheit temperature $F$ .

Evaluate $h open parenthesis 104 close parenthesis$ and explain what it tells us about the temperature.
Find the Fahrenheit temperature of $37 degrees$ Celsius by substituting $C equals 37$ in the formula and solving for $F$ .
Find a formula for the inverse function.
Write your answer to part (b) with inverse function notation.

If you are flying in an airplane at an altitude of $h$ miles, on a clear day you can see a distance of $d$ miles to the horizon, where $d = f (h) = \sqrt{7920 h}$ .

Evaluate $f open parenthesis 0.5 close parenthesis$ and explain what it tells us about the horizon.
Find the altitude needed in order to see a distance of $10$ mile by substituting $d equals 10$ in the formula and solving for $h$ .
Find a formula for the inverse function.
Write your answer to part (b) with inverse function notation.

$f open parenthesis 0.5 close parenthesis approximately equals 62.9$ . At an altitude of $0.5$ miles, you can see $62.9$ miles to the horizon.
$0.0126$ mile, or $66.7$ feet
$h = f^{- 1} (d) = \frac{d^{2}}{7920}$
$f^{- 1} (10) \approx 0.0126$

A moving ship creates waves that impede its own speed. The function $v = f (L) = 1.3 \sqrt{L}$ gives the ship's maximum speed in knots in terms of its length, $L$ , in feet.

Evaluate $f open parenthesis 400 close parenthesis$ and explain what it tells us about the ship's speed.
Find the length needed for a maximum speed of $35$ knots by substituting $v equals 35$ in the formula and solving for $L$ .
Find a formula for the inverse function.
Write your answer to part (b) with inverse function notation.

Use the graph of $h (x) = \sqrt{5 - x}$ to find $h^{- 1} (3)$ .
Find a formula for $h^{- 1} (x)$ and evaluate $h^{- 1} (3)$ .

$h^{- 1} (3) \approx - 4$
$h^{- 1} (x) = 5 - x^{2}$ ; $h^{- 1} (3) = - 4$

Use the graph of $g (x) = \frac{1}{3 - x}$ to find $g^{- 1} (- 2)$ .
Find a formula for $g^{- 1} (x)$ and evaluate $g^{- 1} (- 2)$ .

Find $f^{- 1}$ for the function $f (x) = (x - 2)^{3}$ .
Show that $f^{- 1}$ undoes the effect of $f$ on $x equals 4$ .
Show that $f$ undoes the effect of $f^{- 1}$ on $x = - 8$ .
Graph the function and its inverse on the same grid, along with the graph of $y = x$ .

$f^{- 1} (y) = 3 \sqrt[3]{y} + 2$
$f^{- 1} (f (4)) = f^{- 1} (8) = 4$
$f (f^{- 1} (- 8)) = f (0) = - 8$

Find $f^{- 1}$ for the function $f (x) = \frac{2}{x + 1}$ .
Show that $f^{- 1}$ undoes the effect of $f$ on $x equals 3$ .
Show that $f$ undoes the effect of $f^{- 1}$ on $x = - 1$ .
Graph the function and its inverse on the same grid, along with the graph of $y = x$ .

If $F (t) = \frac{2}{3} t + 1$ , find $F^{- 1} (5)$ .

$6$

If $G (s) = \frac{s - 3}{4}$ , find $G^{- 1} (- 2)$ .

If $m (v) = 6 - \frac{2}{v}$ , find $m^{- 1} (- 3)$ .

$\frac{2}{9}$

If $p (z) = 1 - 2 z^{3}$ , find $p^{- 1} (7)$ .

If $f (x) = \frac{x + 2}{x - 1}$ , find $f^{- 1} (2)$ .

$4$

If $g (n) = \frac{3 n + 1}{n - 3}$ , find $g^{- 1} (- 2)$ .

For Problems 23–26,

Use the graph to make a table of values for the function $y = f (x)$ .
Make a table of values and a graph of the inverse function.

$x$ $0$ $6$
$y$ $300$ $1200$
$x$ $300$ $1200$
$y$ $0$ $6$

$x$	$0$	$6$
$y$	$300$	$1200$

$x$	$300$	$1200$
$y$	$0$	$6$

$x$ $0$ $1$ $2$
$y$ $5$ $20$ $100$
$x$ $5$ $20$ $100$
$y$ $0$ $1$ $2$

$x$	$0$	$1$	$2$
$y$	$5$	$20$	$100$

$x$	$5$	$20$	$100$
$y$	$0$	$1$	$2$

For Problems 27–32,

Find a formula for the inverse of the function.
Graph the function and its inverse on the same set of axes, along with the graph of $y = x$ .

$f (x) = 2 x - 6$

$f^{- 1} (x) = \frac{x + 6}{2}$

$f (x) = 3 x - 1$

$f (x) = x^{3} + 1$

$f^{- 1} (x) = \sqrt[3]{x - 1}$

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

$f (x) = \frac{1}{x - 1}$

$f^{- 1} (x) = \frac{1}{x} + 1$

$f (x) = \frac{1}{x} - 3$

Find the domain and range of the function $g (x) = \sqrt{4 - x}$ .
Find a formula for $g^{- 1} (x)$ .
State the domain and range of $g^{- 1} (x)$ .
Graph $g$ and $g^{- 1}$ on the same grid.

Domain: $open parenthesis minus ∞, 4 ]$ ; Range: $[ 0, ∞ close parenthesis$
$g^{- 1} (x) = 4 - x^{2}$
Domain: $[ 0, ∞ close parenthesis$ ; Range: $open parenthesis minus ∞, 4 ]$

Find the domain and range of the function $g (x) = 8 - \sqrt{x}$ .
Find a formula for $g^{- 1} (x)$ .
State the domain and range of $g^{- 1} (x)$ .
Graph $g$ and $g^{- 1}$ on the same grid.

Which of the functions in Problems 35–42 have inverses that are also functions?

(a) and (d)

$f (x) = x$
$f (x) = x^{2}$

(a)

$f (x) = x^{3}$
$f open parenthesis x close parenthesis equals | x |$

$f (x) = \frac{1}{x}$
$f (x) = \frac{1}{x^{2}}$

(a)

$f (x) = \sqrt{x}$
$f (x) = \sqrt[3]{x}$

$f (x) = 2^{x}$
$f (x) = {(\frac{1}{2})}^{x}$

(a) and (b)

$f (x) = x^{3} + x^{2}$
$f (x) = x^{3} + x$

Find a formula for each function shown in (a)–(d). Then match each function with its inverse from I–IV.

$f (x) = 4 + 2 x$ ; IV
$f (x) = 2 - \frac{x}{2}$ ; III
$f (x) = - 4 - 2 x$ ; I
$f (x) = \frac{x}{2}$ ; II

Find a formula for each function shown in (a)–(d). Then match each function with its inverse from I–IV.

For Problems 45 and 46, use the graph of $f$ to match the other graphs with the appropriate function. (Hint: Look at the coordinates of some specific points.)

$- f$
$\frac{1}{f}$
$f^{- 1}$

$- f$
$\frac{1}{f}$
$f^{- 1}$

Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.