Inverse Functions

LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

  • Verify inverse functions.
  • Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
  • Find or evaluate the inverse of a function.
  • Use the graph of a one-to-one function to graph its inverse function on the same axes.


A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.

If some physical machines can run in two directions, we might ask whether some of the function "machines" we have been studying can also run backwards. Figure 1 provides a visual representation of this question. In this section, we will consider the reverse nature of functions.

Diagram of a function and what would be its inverse. Figure 1. Can a function "machine" operate in reverse?


Verifying That Two Functions Are Inverse Functions

Suppose a fashion designer traveling to Milan for a fashion show wants to know what the temperature will be. He is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, he asks his assistant, Betty, to convert 75 degrees Fahrenheit to degrees Celsius. She finds the formula

C=59(F32)C=\frac{5}{9}\left(F - 32\right)

and substitutes 75 for

FF
to calculate

59(7532)24C\frac{5}{9}\left(75 - 32\right)\approx {24}^{ \circ} {C}
.
A forecast of Monday’s through Thursday’s weather. Figure 2


Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, he sends his assistant the week’s weather forecast for Milan, and asks her to convert all of the temperatures to degrees Fahrenheit.

At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for

FF
after substituting a value for
CC
. For example, to convert 26 degrees Celsius, she could write

{26=59(F32)2695=F32F=2695+3279\begin{cases}26=\frac{5}{9}\left(F - 32\right)\qquad \\ 26\cdot \frac{9}{5}=F - 32\qquad \\ F=26\cdot \frac{9}{5}+32\approx 79\qquad \end{cases}

After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.

The formula for which Betty is searching corresponds to the idea of an inverse function, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.

Given a function

f(x)f\left(x\right)
, we represent its inverse as
f1(x){f}^{-1}\left(x\right)
, read as
"f"f
inverse of
x."x.\text{"}
The raised
1-1
is part of the notation. It is not an exponent; it does not imply a power of
1-1
. In other words,
f1(x){f}^{-1}\left(x\right)
does not mean
1f(x)\frac{1}{f\left(x\right)}
because
1f(x)\frac{1}{f\left(x\right)}
is the reciprocal of
ff
and not the inverse.

The "exponent-like" notation comes from an analogy between function composition and multiplication: just as

a1a=1{a}^{-1}a=1
(1 is the identity element for multiplication) for any nonzero number
aa
, so
f1f{f}^{-1}\circ f
equals the identity function, that is,

(f1f)(x)=f1(f(x))=f1(y)=x\left({f}^{-1}\circ f\right)\left(x\right)={f}^{-1}\left(f\left(x\right)\right)={f}^{-1}\left(y\right)=x

This holds for all

xx
in the domain of
ff
. Informally, this means that inverse functions "undo" each other. However, just as zero does not have a reciprocal, some functions do not have inverses.

Given a function

f(x)f\left(x\right)
, we can verify whether some other function
g(x)g\left(x\right)
is the inverse of
f(x)f\left(x\right)
by checking whether either
g(f(x))=xg\left(f\left(x\right)\right)=x
or
f(g(x))=xf\left(g\left(x\right)\right)=x
is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)

For example,

y=4xy=4x
and
y=14xy=\frac{1}{4}x
are inverse functions.

(f1f)(x)=f1(4x)=14(4x)=x\left({f}^{-1}\circ f\right)\left(x\right)={f}^{-1}\left(4x\right)=\frac{1}{4}\left(4x\right)=x

and

(ff1)(x)=f(14x)=4(14x)=x\left({f}^{}\circ {f}^{-1}\right)\left(x\right)=f\left(\frac{1}{4}x\right)=4\left(\frac{1}{4}x\right)=x

A few coordinate pairs from the graph of the function

y=4xy=4x
are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function
y=14xy=\frac{1}{4}x
are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

A General Note: Inverse Function

For any one-to-one function

f(x)=yf\left(x\right)=y
, a function
f1(x){f}^{-1}\left(x\right)
is an inverse function of
ff
if
f1(y)=x{f}^{-1}\left(y\right)=x
. This can also be written as
f1(f(x))=x{f}^{-1}\left(f\left(x\right)\right)=x
for all
xx
in the domain of
ff
. It also follows that
f(f1(x))=xf\left({f}^{-1}\left(x\right)\right)=x
for all
xx
in the domain of
f1{f}^{-1}
if
f1{f}^{-1}
is the inverse of
ff
.

The notation

f1{f}^{-1}
is read
"f\text{"}f
inverse." Like any other function, we can use any variable name as the input for
f1{f}^{-1}
, so we will often write
f1(x){f}^{-1}\left(x\right)
, which we read as
"f"f
inverse of
x."x."


Keep in mind that

f1(x)1f(x){f}^{-1}\left(x\right)\ne \frac{1}{f\left(x\right)}

and not all functions have inverses.

Example 1: Identifying an Inverse Function for a Given Input-Output Pair

If for a particular one-to-one function

f(2)=4f\left(2\right)=4
and
f(5)=12f\left(5\right)=12
, what are the corresponding input and output values for the inverse function?

Solution

The inverse function reverses the input and output quantities, so if

{cf(2)=4, then f1(4)=2;f(5)=12, then f1(12)=5.\begin{cases}{c}f\left(2\right)=4,\text{ then }{f}^{-1}\left(4\right)=2;\\ f\left(5\right)=12,{\text{ then f}}^{-1}\left(12\right)=5.\end{cases}

Alternatively, if we want to name the inverse function

gg
, then
g(4)=2g\left(4\right)=2
and
g(12)=5g\left(12\right)=5
.

Analysis of the Solution

Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed.

(x,f(x))\left(x,f\left(x\right)\right)
(x,g(x))\left(x,g\left(x\right)\right)
(2,4)\left(2,4\right)
(4,2)\left(4,2\right)
(5,12)\left(5,12\right)
(12,5)\left(12,5\right)

Try It 1

Given that

h1(6)=2{h}^{-1}\left(6\right)=2
, what are the corresponding input and output values of the original function
h?h?

Solution


How To: Given two functions
f(x)f\left(x\right)
and
g(x)g\left(x\right)
, test whether the functions are inverses of each other.

  1. Determine whether
    f(g(x))=xf\left(g\left(x\right)\right)=x
    or
    g(f(x))=xg\left(f\left(x\right)\right)=x
    .
  2. If either statement is true, then both are true, and
    g=f1g={f}^{-1}
    and
    f=g1f={g}^{-1}
    . If either statement is false, then both are false, and
    gf1g\ne {f}^{-1}
    and
    fg1f\ne {g}^{-1}
    .


Example 2: Testing Inverse Relationships Algebraically

If

f(x)=1x+2f\left(x\right)=\frac{1}{x+2}
and
g(x)=1x2g\left(x\right)=\frac{1}{x}-2
, is
g=f1?g={f}^{-1}?

Solution

{g(f(x))=1(1x+2)2=x+22=x\begin{cases} g\left(f\left(x\right)\right)=\frac{1}{\left(\frac{1}{x+2}\right)}{-2 }\qquad\\={ x }+{ 2 } -{ 2 }\qquad\\={ x }\qquad \end{cases}

so

g=f1 and f=g1g={f}^{-1}\text{ and }f={g}^{-1}

This is enough to answer yes to the question, but we can also verify the other formula.

{f(g(x))=11x2+2=11x=x\begin{cases} f\left(g\left(x\right)\right)=\frac{1}{\frac{1}{x}-2+2}\\ =\frac{1}{\frac{1}{x}}\qquad \\ =x\qquad \end{cases}

Analysis of the Solution

Notice the inverse operations are in reverse order of the operations from the original function.

Try It 2

If

f(x)=x34f\left(x\right)={x}^{3}-4
and
g(x)=x+43g\left(x\right)=\sqrt[3]{x+4}
, is
g=f1?g={f}^{-1}?

Solution

Example 3: Determining Inverse Relationships for Power Functions

If

f(x)=x3f\left(x\right)={x}^{3}
(the cube function) and
g(x)=13xg\left(x\right)=\frac{1}{3}x
, is
g=f1?g={f}^{-1}?

Solution

f(g(x))=x327xf\left(g\left(x\right)\right)=\frac{{x}^{3}}{27}\ne x

No, the functions are not inverses.

Analysis of the Solution

The correct inverse to the cube is, of course, the cube root

x3=x13\sqrt[3]{x}={x}^{\frac{1}{3}}
, that is, the one-third is an exponent, not a multiplier.

Try It 3

If

f(x)=(x1)3andg(x)=x3+1f\left(x\right)={\left(x - 1\right)}^{3}\text{and}g\left(x\right)=\sqrt[3]{x}+1
, is
g=f1?g={f}^{-1}?

Solution

Finding Domain and Range of Inverse Functions

The outputs of the function
ff
are the inputs to
f1{f}^{-1}
, so the range of
ff
is also the domain of
f1{f}^{-1}
. Likewise, because the inputs to
ff
are the outputs of
f1{f}^{-1}
, the domain of
ff
is the range of
f1{f}^{-1}
. We can visualize the situation.

Domain and range of a function and its inverse. Figure 3. Domain and range of a function and its inverse


When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of

f(x)=xf\left(x\right)=\sqrt{x}
is
f1(x)=x2{f}^{-1}\left(x\right)={x}^{2}
, because a square "undoes" a square root; but the square is only the inverse of the square root on the domain
[0,)\left[0,\infty \right)
, since that is the range of
f(x)=xf\left(x\right)=\sqrt{x}
.

We can look at this problem from the other side, starting with the square (toolkit quadratic) function

f(x)=x2f\left(x\right)={x}^{2}
. If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and –3. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the "inverse" is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.

In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function

f(x)=x2f\left(x\right)={x}^{2}
with its range limited to
[0,)\left[0,\infty \right)
, which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).

If

f(x)=(x1)2f\left(x\right)={\left(x - 1\right)}^{2}
on
[1,)\left[1,\infty \right)
, then the inverse function is
f1(x)=x+1{f}^{-1}\left(x\right)=\sqrt{x}+1
.

  • The domain of
    ff
    = range of
    f1{f}^{-1}
    =
    [1,)\left[1,\infty \right)
    .
  • The domain of
    f1{f}^{-1}
    = range of
    ff
    =
    [0,)\left[0,\infty \right)
    .


Q & A

Is it possible for a function to have more than one inverse?

No. If two supposedly different functions, say,

gg
and
hh
, both meet the definition of being inverses of another function
ff
, then you can prove that
g=hg=h
. We have just seen that some functions only have inverses if we restrict the domain of the original function. In these cases, there may be more than one way to restrict the domain, leading to different inverses. However, on any one domain, the original function still has only one unique inverse.

A General Note: Domain and Range of Inverse Functions

The range of a function

f(x)f\left(x\right)
is the domain of the inverse function
f1(x){f}^{-1}\left(x\right)
.

The domain of

f(x)f\left(x\right)
is the range of
f1(x){f}^{-1}\left(x\right)
.

How To: Given a function, find the domain and range of its inverse.

  1. If the function is one-to-one, write the range of the original function as the domain of the inverse, and write the domain of the original function as the range of the inverse.
  2. If the domain of the original function needs to be restricted to make it one-to-one, then this restricted domain becomes the range of the inverse function.


Example 4: Finding the Inverses of Toolkit Functions

Identify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on which each function is one-to-one, if any. The toolkit functions are reviewed below. We restrict the domain in such a fashion that the function assumes all y-values exactly once.

Constant Identity Quadratic Cubic Reciprocal
f(x)=cf\left(x\right)=c
f(x)=xf\left(x\right)=x
f(x)=x2f\left(x\right)={x}^{2}
f(x)=x3f\left(x\right)={x}^{3}
f(x)=1xf\left(x\right)=\frac{1}{x}
Reciprocal squared Cube root Square root Absolute value
f(x)=1x2f\left(x\right)=\frac{1}{{x}^{2}}
f(x)=x3f\left(x\right)=\sqrt[3]{x}
f(x)=xf\left(x\right)=\sqrt{x}
f(x)=xf\left(x\right)=|x|

Solution

The constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no meaningful inverse.

The absolute value function can be restricted to the domain

[0,)\left[0,\infty \right)
, where it is equal to the identity function.

The reciprocal-squared function can be restricted to the domain

(0,)\left(0,\infty \right)
.

Analysis of the Solution

We can see that these functions (if unrestricted) are not one-to-one by looking at their graphs. They both would fail the horizontal line test. However, if a function is restricted to a certain domain so that it passes the horizontal line test, then in that restricted domain, it can have an inverse.

Graph of an absolute function. Figure 4. (a) Absolute value (b) Reciprocal squared


Try It 4

The domain of function

ff
is
(1,)\left(1,\infty \right)
and the range of function
ff
is
(,2)\left(\mathrm{-\infty },-2\right)
. Find the domain and range of the inverse function.

Solution

Finding and Evaluating Inverse Functions

Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.

Inverting Tabular Functions

Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.

Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.

Example 5: Interpreting the Inverse of a Tabular Function

A function

f(t)f\left(t\right)
is given below, showing distance in miles that a car has traveled in
tt
minutes. Find and interpret
f1(70){f}^{-1}\left(70\right)
.

t (minutes)t\text{ (minutes)}
30 50 70 90
f(t) (miles)f\left(t\right)\text{ (miles)}
20 40 60 70

Solution

The inverse function takes an output of

ff
and returns an input for
ff
. So in the expression
f1(70){f}^{-1}\left(70\right)
, 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function
ff
, 90 minutes, so
f1(70)=90{f}^{-1}\left(70\right)=90
. The interpretation of this is that, to drive 70 miles, it took 90 minutes.

Alternatively, recall that the definition of the inverse was that if

f(a)=bf\left(a\right)=b
, then
f1(b)=a{f}^{-1}\left(b\right)=a
. By this definition, if we are given
f1(70)=a{f}^{-1}\left(70\right)=a
, then we are looking for a value
aa
so that
f(a)=70f\left(a\right)=70
. In this case, we are looking for a
tt
so that
f(t)=70f\left(t\right)=70
, which is when
t=90t=90
.

Try It 5

Using the table below, find and interpret (a)

 f(60)\text{ }f\left(60\right)
, and (b)
 f1(60)\text{ }{f}^{-1}\left(60\right)
.

t (minutes)t\text{ (minutes)}
30 50 60 70 90
f(t) (miles)f\left(t\right)\text{ (miles)}
20 40 50 60 70
Solution

Evaluating the Inverse of a Function, Given a Graph of the Original Function

We saw in Functions and Function Notation that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.

How To: Given the graph of a function, evaluate its inverse at specific points.

  1. Find the desired input on the y-axis of the given graph.
  2. Read the inverse function’s output from the x-axis of the given graph.


Example 6: Evaluating a Function and Its Inverse from a Graph at Specific Points

A function
g(x)g\left(x\right)
is given in Figure 5. Find
g(3)g\left(3\right)
and
g1(3){g}^{-1}\left(3\right)
.

Graph of g(x). Figure 5


Solution

To evaluate

g(3)g\left(3\right)
, we find 3 on the x-axis and find the corresponding output value on the y-axis. The point
(3,1)\left(3,1\right)
tells us that
g(3)=1g\left(3\right)=1
.

To evaluate
g1(3){g}^{-1}\left(3\right)
, recall that by definition
g1(3){g}^{-1}\left(3\right)
means the value of x for which
g(x)=3g\left(x\right)=3
. By looking for the output value 3 on the vertical axis, we find the point
(5,3)\left(5,3\right)
on the graph, which means
g(5)=3g\left(5\right)=3
, so by definition,
g1(3)=5{g}^{-1}\left(3\right)=5
.

Graph of g(x). Figure 6


Try It 6

Using the graph in Example 6, (a) find

g1(1){g}^{-1}\left(1\right)
, and (b) estimate
g1(4){g}^{-1}\left(4\right)
.

Solution

Finding Inverses of Functions Represented by Formulas

Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula— for example,

yy
as a function of
x-x\text{-\hspace{0.17em}}
we can often find the inverse function by solving to obtain
xx
as a function of
yy
.

How To: Given a function represented by a formula, find the inverse.

  1. Make sure
    ff
    is a one-to-one function.
  2. Solve for
    xx
    .
  3. Interchange
    xx
    and
    yy
    .


Example 7: Inverting the Fahrenheit-to-Celsius Function

Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.

C=59(F32)C=\frac{5}{9}\left(F - 32\right)

Solution

{C=59(F32)C95=F32F=95C+32\begin{cases}\qquad{ C }=\frac{5}{9}\left(F - 32\right)\qquad \\ C\cdot \frac{9}{5}=F - 32\qquad \\ F=\frac{9}{5}C+32\qquad \end{cases}

By solving in general, we have uncovered the inverse function. If

C=h(F)=59(F32)C=h\left(F\right)=\frac{5}{9}\left(F - 32\right)
,

then

F=h1(C)=95C+32F={h}^{-1}\left(C\right)=\frac{9}{5}C+32
.

In this case, we introduced a function

hh
to represent the conversion because the input and output variables are descriptive, and writing
C1{C}^{-1}
could get confusing.

Try It 7

Solve for

xx
in terms of
yy
given
y=13(x5)y=\frac{1}{3}\left(x - 5\right)

Solution

Example 8: Solving to Find an Inverse Function

Find the inverse of the function

f(x)=2x3+4f\left(x\right)=\frac{2}{x - 3}+4
.

Solution

{y=2x3+4Set up an equation.y4=2x3Subtract 4 from both sides.x3=2y4Multiply both sides by x3 and divide by y4.x=2y4+3Add 3 to both sides.\begin{cases}y=\frac{2}{x - 3}+4\qquad & \text{Set up an equation}.\qquad \\ y - 4=\frac{2}{x - 3}\qquad & \text{Subtract 4 from both sides}.\qquad \\ x - 3=\frac{2}{y - 4}\qquad & \text{Multiply both sides by }x - 3\text{ and divide by }y - 4.\qquad \\ x=\frac{2}{y - 4}+3\qquad & \text{Add 3 to both sides}.\qquad \end{cases}

So

f1(y)=2y4+3{f}^{-1}\left(y\right)=\frac{2}{y - 4}+3
or
f1(x)=2x4+3{f}^{-1}\left(x\right)=\frac{2}{x - 4}+3
.

Analysis of the Solution

The domain and range of

ff
exclude the values 3 and 4, respectively.
ff
and
f1{f}^{-1}
are equal at two points but are not the same function, as we can see by creating the table below.

xx
1 2 5
f1(y){f}^{-1}\left(y\right)
f(x)f\left(x\right)
3 2 5
yy

Example 9: Solving to Find an Inverse with Radicals

Find the inverse of the function

f(x)=2+x4f\left(x\right)=2+\sqrt{x - 4}
.

Solution

{y=2+x4(y2)2=x4x=(y2)2+4\begin{cases}y=2+\sqrt{x - 4}\qquad \\ {\left(y - 2\right)}^{2}=x - 4\qquad \\ x={\left(y - 2\right)}^{2}+4\qquad \end{cases}

So

f1(x)=(x2)2+4{f}^{-1}\left(x\right)={\left(x - 2\right)}^{2}+4
.

The domain of

ff
is
[4,)\left[4,\infty \right)
. Notice that the range of
ff
is
[2,)\left[2,\infty \right)
, so this means that the domain of the inverse function
f1{f}^{-1}
is also
[2,)\left[2,\infty \right)
.

Analysis of the Solution

The formula we found for

f1(x){f}^{-1}\left(x\right)
looks like it would be valid for all real
xx
. However,
f1{f}^{-1}
itself must have an inverse (namely,
ff
) so we have to restrict the domain of
f1{f}^{-1}
to
[2,)\left[2,\infty \right)
in order to make
f1{f}^{-1}
a one-to-one function. This domain of
f1{f}^{-1}
is exactly the range of
ff
.

Try It 8

What is the inverse of the function

f(x)=2x?f\left(x\right)=2-\sqrt{x}?
State the domains of both the function and the inverse function.

Solution
Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function
f(x)=x2f\left(x\right)={x}^{2}
restricted to the domain
[0,)\left[0,\infty \right)
, on which this function is one-to-one, and graph it as in Figure 7.

Graph of f(x). Figure 7. Quadratic function with domain restricted to [0, ∞).


Restricting the domain to

[0,)\left[0,\infty \right)
makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.

We already know that the inverse of the toolkit quadratic function is the square root function, that is,

f1(x)=x{f}^{-1}\left(x\right)=\sqrt{x}
. What happens if we graph both
f f\text{ }
and
f1{f}^{-1}
on the same set of axes, using the
x-x\text{-}
axis for the input to both
f and f1?f\text{ and }{f}^{-1}?

We notice a distinct relationship: The graph of
f1(x){f}^{-1}\left(x\right)
is the graph of
f(x)f\left(x\right)
reflected about the diagonal line
y=xy=x
, which we will call the identity line, shown in Figure 8.

Graph of f(x) and f^(-1)(x). Figure 8. Square and square-root functions on the non-negative domain


This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.

Example 10: Finding the Inverse of a Function Using Reflection about the Identity Line

Given the graph of
f(x)f\left(x\right)
, sketch a graph of
f1(x){f}^{-1}\left(x\right)
.

Graph of f^(-1)(x). Figure 9


Solution

This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of

(0,)\left(0,\infty \right)
and range of
(,)\left(-\infty ,\infty \right)
, so the inverse will have a domain of
(,)\left(-\infty ,\infty \right)
and range of
(0,)\left(0,\infty \right)
.

If we reflect this graph over the line
y=xy=x
, the point
(1,0)\left(1,0\right)
reflects to
(0,1)\left(0,1\right)
and the point
(4,2)\left(4,2\right)
reflects to
(2,4)\left(2,4\right)
. Sketching the inverse on the same axes as the original graph gives us the result in Figure 10.

Graph of f(x) and f^(-1)(x). Figure 10. The function and its inverse, showing reflection about the identity line


Try It 9

Draw graphs of the functions

f f\text{ }
and
 f1\text{ }{f}^{-1}
.

Solution

Q & A

Is there any function that is equal to its own inverse?

Yes. If

f=f1f={f}^{-1}
, then
f(f(x))=xf\left(f\left(x\right)\right)=x
, and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because

11x=x\frac{1}{\frac{1}{x}}=x

Any function

f(x)=cxf\left(x\right)=c-x
, where
cc
is a constant, is also equal to its own inverse.

Key Concepts

  • If
    g(x)g\left(x\right)
    is the inverse of
    f(x)f\left(x\right)
    , then
  • g(f(x))=f(g(x))=xg\left(f\left(x\right)\right)=f\left(g\left(x\right)\right)=x
    .
  • Each of the toolkit functions has an inverse.
  • For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
  • A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
  • For a tabular function, exchange the input and output rows to obtain the inverse.
  • The inverse of a function can be determined at specific points on its graph.
  • To find the inverse of a formula, solve the equation
    y=f(x)y=f\left(x\right)
    for
    xx
    as a function of 
    yy
    . Then exchange the labels
    xx
    and
    yy
    .
  • The graph of an inverse function is the reflection of the graph of the original function across the line
    y=xy=x
    .


Glossary

inverse function
for any one-to-one function
f(x)f\left(x\right)
, the inverse is a function
f1(x){f}^{-1}\left(x\right)
such that
f1(f(x))=x{f}^{-1}\left(f\left(x\right)\right)=x
for all
xx
in the domain of
ff
; this also implies that
f(f1(x))=xf\left({f}^{-1}\left(x\right)\right)=x
for all
xx
in the domain of
f1{f}^{-1}


Problem Set

1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

2. Why do we restrict the domain of the function
f(x)=x2f\left(x\right)={x}^{2}
to find the function’s inverse?

3. Can a function be its own inverse? Explain.

4. Are one-to-one functions either always increasing or always decreasing? Why or why not?

5. How do you find the inverse of a function algebraically?

6. Show that the function
f(x)=axf\left(x\right)=a-x
is its own inverse for all real numbers
aa
.

For the following exercises, find
f1(x){f}^{-1}\left(x\right)
for each function.

7.
f(x)=x+3f\left(x\right)=x+3


8. 
f(x)=x+5f\left(x\right)=x+5


9.
f(x)=2xf\left(x\right)=2-x


10. 
f(x)=3xf\left(x\right)=3-x


11. 
f(x)=xx+2f\left(x\right)=\frac{x}{x+2}


12. 
f(x)=2x+35x+4f\left(x\right)=\frac{2x+3}{5x+4}


For the following exercises, find a domain on which each function
ff
is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of
ff
restricted to that domain.

13.
f(x)=(x+7)2f\left(x\right)={\left(x+7\right)}^{2}


14. 
f(x)=(x6)2f\left(x\right)={\left(x - 6\right)}^{2}


15.
f(x)=x25f\left(x\right)={x}^{2}-5


16. Given
f(x)=x2+xf\left(x\right)=\frac{x}{2}+x
and
g(x)=2x1xg\left(x\right)=\frac{2x}{1-x}


a. Find
f(g(x))f\left(g\left(x\right)\right)
and
g(f(x))g\left(f\left(x\right)\right)


b. What does the answer tell us about the relationship between
f(x)f\left(x\right)
and
g(x)?g\left(x\right)?


For the following exercises, use function composition to verify that
f(x)f\left(x\right)
and
g(x)g\left(x\right)
are inverse functions.

17.
f(x)=x13f\left(x\right)=\sqrt[3]{x - 1}
and
g(x)=x3+1g\left(x\right)={x}^{3}+1


18. 
f(x)=3x+5f\left(x\right)=-3x+5
and
g(x)=x53g\left(x\right)=\frac{x - 5}{-3}


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

19.
f(x)=xf\left(x\right)=\sqrt{x}


20. 
f(x)=3x+13f\left(x\right)=\sqrt[3]{3x+1}


21.
f(x)=5x+1f\left(x\right)=-5x+1


22. 
f(x)=x327f\left(x\right)={x}^{3}-27


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

23.

Graph of an upright parabola with vertex at (10, -10), passing through (0,0) and (25,0) 24.

Graph of a step-function, with y = -5 for {x|-10<=x,0} and y = 0 for {x|0<=x<10} For the following exercises, use the graph of
ff
shown in [link].

Graph of the line y = (-3/2)x + 3 25. Find
f(0)f\left(0\right)
.

26. Solve
f(x)=0f\left(x\right)=0
.

27. Find
f1(0){f}^{-1}\left(0\right)
.

28. Solve
f1(x)=0{f}^{-1}\left(x\right)=0
.

For the following exercises, use the graph of the one-to-one function shown below.

Graph of a square root function for {x|x>=2} 29. Sketch the graph of
f1{f}^{-1}
.

30. Find
f(6) and f1(2)f\left(6\right)\text{ and }{f}^{-1}\left(2\right)
.

31. If the complete graph of
ff
is shown, find the domain of
ff
.

32. If the complete graph of
ff
is shown, find the range of
ff
.

For the following exercises, evaluate or solve, assuming that the function
ff
is one-to-one.

33. If
f(6)=7f\left(6\right)=7
, find
f1(7){f}^{-1}\left(7\right)
.

34. If
f(3)=2f\left(3\right)=2
, find
f1(2){f}^{-1}\left(2\right)
.

35. If
f1(4)=8{f}^{-1}\left(-4\right)=-8
, find
f(8)f\left(-8\right)
.

36. If
f1(2)=1{f}^{-1}\left(-2\right)=-1
, find
f(1)f\left(-1\right)
.

For the following exercises, use the values listed in the table below to evaluate or solve.

xx
f(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
37. Find
f(1)f\left(1\right)
.

38. Solve
f(x)=3f\left(x\right)=3
.

39. Find
f1(0){f}^{-1}\left(0\right)
.

40. Solve
f1(x)=7{f}^{-1}\left(x\right)=7
.

41. Use the tabular representation of
ff
to create a table for
f1(x){f}^{-1}\left(x\right)
.

xx
3 6 9 13 14
f(x)f\left(x\right)
1 4 7 12 16
For the following exercises, find the inverse function. Then, graph the function and its inverse.

42.
f(x)=3x2f\left(x\right)=\frac{3}{x - 2}


43.
f(x)=x31f\left(x\right)={x}^{3}-1


44. Find the inverse function of
f(x)=1x1f\left(x\right)=\frac{1}{x - 1}
. Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

45. To convert from
xx
degrees Celsius to
yy
degrees Fahrenheit, we use the formula
f(x)=95x+32f\left(x\right)=\frac{9}{5}x+32
. Find the inverse function, if it exists, and explain its meaning.

46. The circumference
CC
of a circle is a function of its radius given by
C(r)=2πrC\left(r\right)=2\pi r
. Express the radius of a circle as a function of its circumference. Call this function
r(C)r\left(C\right)
. Find
r(36π)r\left(36\pi \right)
and interpret its meaning.

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

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