Functions and Their Graphs

1.5	Inverse Functions

LEARNING OBJECTIVES

By the end of this section you should be able to


■	determine which functions have inverses;

■	find a formula for an inverse function (when possible);

■	use the composition of a function and its inverse to check that an inverse function has been correctly found;

■	find the domain and range of an inverse function.

The Inverse Problem

The concept of an inverse function will play a key role in this book in defining roots, logarithms, and inverse trigonometric functions. To motivate this concept, we begin with some simple examples.

Suppose

is the function defined by

. Given a value of

, we can find the value of

by using the formula defining

. For example, taking

, we see that

equals

In the inverse problem, we are given the value of

and asked to find the value of

. The following example illustrates the idea of the inverse problem:

EXAMPLE 1

Suppose

is the function defined by

(a)	Find such that .

(b)	Find such that .

(c)

For each number

, find a number

such that

SOLUTION


(a)	Solving the equation for , we get .

(b)	Solving the equation for , we get .

(c)	Solving the equation for , we get .

For each number

, part (c) of the example above asks for the number

such that

. That number

is called

(pronounced “

inverse of

”). The example above shows that if

, then

and

and, more generally,

for every number

Inverse functions will be defined more precisely after we work through some examples.

To see how inverse functions can arise in real-world problems, suppose you know that a temperature of

degrees Celsius corresponds to

degrees Fahrenheit (we will derive this formula in Example 5 in Section 2.1). In other words, you know that the function

that converts the Celsius temperature scale to the Fahrenheit temperature scale is given by the formula

For example, because

, this formula shows that

degrees Celsius corresponds to

degrees Fahrenheit.

If you are given a temperature on the Fahrenheit scale and asked to convert it to Celsius, then you are facing the problem of finding the inverse of the function above, as shown in the following example.

EXAMPLE 2

(a)

Convert

degrees Fahrenheit to the Celsius scale.

The Fahrenheit temperature scale was invented in the

century by the German physicist and engineer Daniel Gabriel Fahrenheit.

(b)

For each temperature

on the Fahrenheit scale, what is the corresponding temperature on the Celsius scale?

SOLUTION

Let

Thus

degrees Celsius corresponds to

degrees Fahrenheit.

The Celsius temperature scale is named in honor of the

century Swedish astronomer Anders Celsius, who originally proposed a temperature scale with

as the boiling point of water and

as the freezing point. Later this was reversed, giving us the familiar scale in which higher numbers correspond to hotter temperatures.

(a)

We need to find

such that

. Solving the equation

for

, we get

. Thus

degrees Celsius corresponds to

degrees Fahrenheit.

(b)

For each number

, we need to find

such that

. Solving the equation

for

, we get

Thus

degrees Celsius corresponds to

degrees Fahrenheit.

In the example above we have

. For each number

, part (b) of the example above asks for the number

such that

. We call that number

. Part (a) of the example above shows that

; part (b) shows more generally that

In this example, the function

converts from Celsius to Fahrenheit, and the function

goes in the other direction, converting from Fahrenheit to Celsius.

One-to-one Functions

To see the difficulties that can arise with inverse problems, consider the function

, with domain the set of real numbers, defined by the formula

Suppose we are told that

is a number such that

, and we are asked to find the value of

. Of course

, but also

. Thus with the information given we have no way to determine a unique value of

such that

. Hence in this case an inverse function does not exist.

The difficulty with the lack of a unique solution to an inverse problem can often be fixed by changing the domain. For example, consider the function

, with domain the set of positive numbers, defined by the formula

Note that

is defined by the same formula as

in the previous paragraph, but these two functions are not the same because they have different domains. Now if we are told that

is a number in the domain of

such that

and we are asked to find

, we can assert that

. More generally, given any positive number

, we can ask for the number

in the domain of

such that

. This number

, which depends on

, is denoted

, and is given by the formula

We saw earlier that the function

defined by

(and with domain equal to the set of real numbers) does not have an inverse because, in particular, the equation

has more than one solution. A function is called one-to-one if this situation does not arise.

One-to-one

A function

is called one-to-one if for each number

in the range of

there is exactly one number

in the domain of

such that

Functions that are one-to-one are precisely the functions that have inverses.

For example, the function

, with domain the set of real numbers, defined by

is not one-to-one because there are two distinct numbers

in the domain of

such that

(we could have used any positive number instead of

to show that

is not one-to-one). In contrast, the function

, with domain the set of positive numbers, defined by

is one-to-one.

The Definition of an Inverse Function

We are now ready to give the formal definition of an inverse function.

Definition of f ⁻¹

Suppose

is a one-to-one function.


•	If is in the range of , then is defined to be the number such that .

•	The function is called the inverse function of .

Short version:


•	means .

EXAMPLE 3

Suppose

(a)	Evaluate .

(b)

Find a formula for

SOLUTION

(a)

To evaluate

, we must find the number

such that

. In other words, we must solve the equation

. The solution to this equation is

. Thus

, and hence

(b)

Fix a number

. To find a formula for

, we must find the number

such that

. In other words, we must solve the equation

for

. The solution to this equation is

. Thus

, and hence

Video Clip - Section 1.5, Example 3: Finding the Inverse Function

is a one-to-one function, then for each

in the range of

we have a uniquely defined number

. Thus

is itself a function.

The inverse function is not defined for a function that is not one-to-one.

Think of

as undoing whatever

does. This list gives some examples of a function

and its inverse

The first entry in the list above shows that if

is the function that adds

to a number, then

is the function that subtracts

from a number.

The second entry in the list above shows that if

is the function that multiplies a number by

, then

is the function that divides a number by

Similarly, the third entry in the list above shows that if

is the function that squares a number, then

is the function that takes the square root of a number (here the domain of

is assumed to be the nonnegative numbers, so that we have a one-to-one function).

Finally, the fourth entry in the list above shows that if

is the function that takes the square root of a number, then

is the function that squares a number (here the domain of

is assumed to be the nonnegative numbers, because the square root of a negative number is not defined as a real number).

In Section 1.1 we saw that a function

can be thought of as a machine that takes an input

and produces an output

. Similarly, we can think of

as a machine that takes an input

and produces an output

Thought of as a machine,

reverses the action of

The procedure for finding a formula for an inverse function can be described as follows:

Finding a formula for an inverse function

Suppose

is a one-to-one function. To find a formula for

, solve the equation

for

in terms of

EXAMPLE 4

Suppose

for every

. Find a formula for

SOLUTION

To find a formula for

, we need to solve the equation

for

in terms of

. This can be done be multiplying both sides of the equation above by

, getting

which can then be rewritten as

which can then be solved for

, getting

Thus

for every

The equation

has no solution (try to solve it to see why), and thus

is not defined.

The Domain and Range of an Inverse Function

The domain and range of a one-to-one function are nicely related to the domain and range of its inverse. To understand this relationship, consider a one-to-one function

. Note that

is defined precisely when

is in the range of

. Thus the domain of

equals the range of

Similarly, because

reverses the action of

, a moment's thought shows that the range of

equals the domain of

. We can summarize the relationship between the domains and ranges of functions and their inverses as follows:

Domain and range of an inverse function

is a one-to-one function, then


•	the domain of equals the range of ;

•	the range of equals the domain of .

EXAMPLE 5

Suppose the domain of

is the interval

, with

defined on this domain by the equation

(a)	What is the range of ?

(b)	Find a formula for the inverse function .

(c)	What is the domain of the inverse function ?

(d)

What is the range of the inverse function

SOLUTION

(a)

The range of

is the interval

because that interval is equal to the set of squares of numbers in the interval

(b)

Suppose

is in the range of

, which is the interval

. To find a formula for

, we have to solve for

in the equation

. In other words, we have to solve the equation

for

. The solution

must be in the domain of

, which is

, and in particular

must be nonnegative. Thus we have

. In other words,

(c)

The domain of the inverse function

is the interval

, which is the range of

(d)

The range of the inverse function

is the interval

, which is the domain of

This example illustrates how the inverse function interchanges the domain and range of the original function.

The Composition of a Function and Its Inverse

The following example will help motivate our next result.

EXAMPLE 6

Suppose

is the function whose domain is the set of real numbers, with

defined by

(a)	Find a formula for .

(b)

Find a formula for

SOLUTION

As we saw in Example 3,

. Thus we have the following:


(a)

(b)

Similar equations hold for the composition of any one-to-one function and its inverse:

The composition of a function and its inverse

Suppose

is a one-to-one function. Then


•	for every in the range of ;

•	for every in the domain of .

To see why these results hold, first suppose

is a number in the range of

. Let

. Then

. Thus

as claimed above.

To verify the second conclusion in the box above, suppose

is a number in the domain of

. Let

. Then

. Thus

as claimed.

Recall that

is the identity function defined by

(where we have left the domain vague), or we could equally well define

by the equation

. The results in the box above could be expressed by the equations

Here the

in the first equation above has domain equal to the range of

(which equals the domain of

), and the

in the second equation above has the same domain as

. The equations above explain why the terminology “inverse” is used for the inverse function:

is the inverse of

under composition in the sense that the composition of

and

in either order gives the identity function.

The figure below illustrates the equation

, thinking of

and

as machines.

Here we start with

as input and end with

as output. Thus this figure illustrates the equation

We start with

as the input. The first machine produces output

, which then becomes the input for the second machine. When

is input into the second machine, the output is

because the second machine, which is based on

, reverses the action of

Suppose you need to compute the inverse of a function

. As discussed earlier, to find a formula for

you need to solve the equation

for

in terms of

. Once you have obtained a formula for

, a good way to check your result is to verify one or both of the equations in the box above.

EXAMPLE 7

Suppose

which is the formula for converting the Celsius temperature scale to the Fahrenheit scale. We computed earlier that the inverse to this function is given by the formula

Check that this formula is correct by verifying that

for every real number

SOLUTION

To check that we have the right formula for

, we compute as follows:

To be doubly safe that we are not making an algebraic manipulation error, we could also verify that

for every real number

. However, one check is usually good enough.

Thus

, which means that our formula for

is correct. If our computation of

had simplified to anything other than

, we would know that we had made a mistake in computing

Comments About Notation

The notation

leads naturally to the notation

. Recall, however, that in defining a function the variable is simply a placeholder. Thus we could use other letters, including

, as the variable for the inverse function. For example, consider the function

, with domain equal to the set of positive numbers, defined by the equation

As we know, the inverse function is given by the formula

However, the inverse function could also be characterized by the formula

Other letters could also be used as the placeholder. For example, we might also characterize the inverse function by the formula

The notation

for the inverse of a function (which means the inverse under composition) should not be confused with the multiplicative inverse

. In other words,

. However, if the exponent

is placed anywhere other than immediately after a function symbol, then it should probably be interpreted as a multiplicative inverse.

Do not confuse

with

EXAMPLE 8

Suppose

, with the domain of

being the set of positive numbers.

(a)	Evaluate .

(b)	Evaluate .

(c)

Evaluate

SOLUTION

(a)

To evaluate

, we must find a positive number

such that

. In other words, we must solve the equation

. The solution to this equation is

. Thus

, and hence

(b)

(c)

Video Clip - Section 1.5, Example 8: Notation for Inverse Function

When dealing with real-world problems, you may want to choose the notation to reflect the context. The next example illustrates this idea, with the use of the variable

to denote distance and

to denote time.

EXAMPLE 9

Suppose you ran a marathon (26.2 miles) in exactly 4 hours. Let

be the function with domain

such that

is the number of minutes since the start of the race at which you reached distance

miles from the starting line.

(a)	What is the range of ?

(b)	What is the domain of the inverse function ?

(c)

What is the meaning of

for a number

in the domain of

SOLUTION


(a)	Because 4 hours equals 240 minutes, the range of is the interval .

(b)	As usual, the domain of is the range of . Thus the domain of is the interval .

(c)	The function reverses the roles of the input and the output as compared to the function . Thus is the distance in miles you had run from the starting line at time minutes after the start of the race.

Video Clip - Section 1.5, Example 9: Domain of an Inverse Function

	EXERCISES

For Exercises 1-8, check your answer by evaluating the appropriate function at your answer.

Suppose

. Evaluate

2	Suppose . Evaluate .

Suppose

. Evaluate

SOLUTION

We need to find a number

such that

. In other words, we need to solve the equation

Multiplying both sides of this equation by

gives the equation

which has solution

. Thus

CHECK To check that

, we need to verify that

. We have

as desired.

4	Suppose . Evaluate .

Suppose

. Find a formula for

SOLUTION

For each number

, we need to find a number

such that

. In other words, we need to solve the equation

for

in terms of

. Subtracting

from both sides of the equation above and then dividing both sides by

gives

Thus

for every number

CHECK To check that

, we need to verify that

. We have

as desired.

6	Suppose . Find a formula for .

Suppose

. Find a formula for

SOLUTION

For each number

, we need to find a number

such that

. In other words, we need to solve the equation

for

in terms of

. Multiplying both sides of this equation by

and then collecting all the terms with

on one side gives

Rewriting the left side as

and then dividing both sides by

gives

Thus

for every number

CHECK To check that

, we need to verify that

. We have

Multiplying numerator and denominator of the expression on the right by

gives

as desired.

8	Suppose . Find a formula for .

Suppose

(a)	Evaluate .

(b)	Evaluate .

(c)

Evaluate

SOLUTION

(a)

We need to find a number

such that

. In other words, we need to solve the equation

Subtracting

from both sides and then multiplying both sides by

gives the equation

which has solution

. Thus

(b)

Note that

Thus

(c)

Suppose

(a)	Evaluate .

(b)	Evaluate .

(c)	Evaluate .

Suppose

, with the domain of

being the set of positive numbers. Evaluate

SOLUTION

We need to find a positive number

such that

. In other words, we need to find a positive solution to the equation

which is equivalent to the equation

The equation above has solutions

Because the domain of

is the set of positive numbers, the value of

that we seek must be positive. The second solution above is negative; thus it can be discarded, giving

12	Suppose , with the domain of being the set of positive numbers. Evaluate .

Suppose

, where the domain of

is the set of positive numbers. Find a formula for

SOLUTION

For each number

, we need to find a number

such that

. In other words, we need to solve the equation

for

in terms of

. Subtracting

from both sides of the equation above and then dividing both sides by

and taking square roots gives

where we chose the positive square root because

is required to be a positive number. Thus

for every number

(the restriction that

is required to ensure that we get a positive number when evaluating the formula above).

14	Suppose , where the domain of is the set of positive numbers. Find a formula for .

For each of the functions

given in Exercises 15-24:


(a)	Find the domain of .

(b)	Find the range of .

(c)	Find a formula for .

(d)	Find the domain of .

(e)	Find the range of .

You can check your solutions to part (c) by verifying that

and

(recall that

is the function defined by

SOLUTION

(a)

The expression

makes sense for all real numbers

. Thus the domain of

is the set of real numbers.

(b)

To find the range of

, we need to find the numbers

such that

for some

in the domain of

. In other words, we need to find the values of

such that the equation above can be solved for a real number

. Solving the equation above for

, we get

The expression above on the right makes sense for every real number

. Thus the range of

is the set of real numbers.

(c)

The expression above shows that

is given by the formula

(d)

The domain of

equals the range of

. Thus the domain of

is the set of real numbers.

(e)

The range of

equals the domain of

. Thus the range of

is the set of real numbers.

SOLUTION

(a)

The expression

makes sense except when

. Solving this equation for

gives

. Thus the domain of

is the set

(b)

To find the range of

, we need to find the numbers

such that

for some

in the domain of

. In other words, we need to find the values of

such that the equation above can be solved for a real number

. To solve this equation for

, multiply both sides by

, getting

Now subtract

from both sides, then divide by

, getting

The expression above on the right makes sense for every real number

and produces a number

(because the equation

leads to nonsense, as you can verify if you try to solve it for

). Thus the range of

is the set

(c)

The expression above shows that

is given by the formula

(d)

The domain of

equals the range of

. Thus the domain of

is the set

(e)

The range of

equals the domain of

. Thus the range of

is the set

SOLUTION

(a)

The expression

makes sense except when

. Thus the domain of

is the set

(b)

To find the range of

, we need to find the numbers

such that

for some

in the domain of

. In other words, we need to find the values of

such that the equation above can be solved for a real number

. To solve this equation for

, multiply both sides by

, getting

Now subtract

from both sides, getting

Dividing by

gives

The expression above on the right makes sense for every real number

and produces a number

(because the equation

leads to nonsense, as you can verify if you try to solve it for

). Thus the range of

is the set

(c)

The expression above shows that

is given by the formula

(d)

The domain of

equals the range of

. Thus the domain of

is the set

(e)

The range of

equals the domain of

. Thus the range of

is the set

SOLUTION

(a)

The expression defining

makes sense for all real numbers

. Thus the domain of

is the set of real numbers.

(b)

To find the range of

, we need to find the numbers

such that

for some real number

. From the definition of

, we see that if

, then

, and if

, then

. Thus every real number

is in the range of

. In other words, the range of

is the set of real numbers.

(c)

From the paragraph above, we see that

is given by the formula

(d)

The domain of

equals the range of

. Thus the domain of

is the set of real numbers.

(e)

The range of

equals the domain of

. Thus the range of

is the set of real numbers.

, where the domain of

equals

SOLUTION

(a)

As part of the definition of the function

, the domain has been specified to be the interval

, which is the set of positive numbers.

(b)

To find the range of

, we need to find the numbers

such that

for some

in the domain of

. In other words, we need to find the values of

such that the equation above can be solved for a positive number

. To solve this equation for

, subtract

from both sides and then take square roots of both sides, getting

where we chose the positive square root of

because

is required to be a positive number.

The expression above on the right makes sense and produces a positive number

for every number

. Thus the range of

is the interval

(c)

The expression above shows that

is given by the formula

(d)

The domain of

equals the range of

. Thus the domain of

is the interval

(e)

The range of

equals the domain of

. Thus the range of

is the interval

, which is the set of positive numbers.

24	, where the domain of equals .

Suppose

. Which of the numbers listed below equals

[For this particular function, it is not possible to find a formula for

SOLUTION

First we test whether or not

equals

by checking whether or not

equals

. Using a calculator, we find that

which means that

Next we test whether or not

equals

by checking whether or not

equals

. Using a calculator, we find that

which means that

Next we test whether or not

equals

by checking whether or not

equals

. Using a calculator, we find that

which means that

Suppose

. Which of the numbers listed below equals

[For this particular function, it is not possible to find a formula for

For Exercises 27-28, use the U. S. 2011 federal income tax function for a single person as defined in Example 2 of Section 1.1.

What is the taxable income of a single person who paid $10,000 in federal taxes for 2011?

28	What is the taxable income of a single person who paid $20,000 in federal taxes for 2011?

Suppose

. Evaluate

Suppose

. Evaluate

	PROBLEMS

The exact number of meters in

yards is

, where

is the function defined by

(a)	Find a formula for .

(b)	What is the meaning of ?

The exact number of kilometers in

miles is

, where

is the function defined by

(a)	Find a formula for .

(b)	What is the meaning of ?

A temperature

degrees Fahrenheit corresponds to

degrees on the Kelvin temperature scale, where

(a)	Find a formula for .

(b)	What is the meaning of ?

(c)	Evaluate . (This is absolute zero, the lowest possible temperature, because all molecular activity stops at degrees Kelvin.)

34	Suppose is the federal income tax function given by Example 2 of Section 1.1. What is the meaning of the function ?

Suppose

is the function whose domain is the set of real numbers, with

defined on this domain by the formula

Explain why

is not a one-to-one function.

Suppose

is the function whose domain is the interval

, with

defined on this domain by the formula

Explain why

is not a one-to-one function.

37	Show that if is the function defined by , where , then is a one-to-one function.

38	Show that if is the function defined by , where , then the inverse function is defined by the formula .

Consider the function

whose domain is the interval

, with

defined on this domain by the formula

Does

have an inverse? If so, find it, along with its domain and range. If not, explain why not.

Consider the function

whose domain is the interval

, with

defined on this domain by the formula

Does

have an inverse? If so, find it, along with its domain and range. If not, explain why not.

Suppose

is a one-to-one function. Explain why the inverse of the inverse of

equals

. In other words, explain why

The function

defined by

is one-to-one (here the domain of

is the set of real numbers). Compute

for four different values of

of your choice.

[For this particular function, it is not possible to find a formula for

43	Suppose is a function whose domain equals and whose range equals . Explain why is a one-to-one function.

44	Suppose is a function whose domain equals and whose range equals . Explain why is not a one-to-one function.

45	Show that the composition of two one-to-one functions is a one-to-one function.

46	Give an example to show that the sum of two one-to-one functions is not necessarily a one-to-one function.

47	Give an example to show that the product of two one-to-one functions is not necessarily a one-to-one function.

48	Give an example of a function such that the domain of and the range of both equal the set of integers, but is not a one-to-one function.

49	Give an example of a one-to-one function whose domain equals the set of integers and whose range equals the set of positive integers.