# 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.## 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

and substitutes 75 for

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

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

*not*mean

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

This holds for all

**reciprocal**, some functions do not have inverses.

Given a function

For example,

and

A few coordinate pairs from the graph of the function

### A General Note: Inverse Function

For any **one-to-one function**

**inverse function**of

The notation

Keep in mind that

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

### Solution

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

Alternatively, if we want to name the inverse function

### Try It 1

Given that

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

- Determine whether $f\left(g\left(x\right)\right)=x$or$g\left(f\left(x\right)\right)=x$.
- If either statement is true, then both are true, and $g={f}^{-1}$and$f={g}^{-1}$. If either statement is false, then both are false, and$g\ne {f}^{-1}$and$f\ne {g}^{-1}$.

### Example 2: Testing Inverse Relationships Algebraically

If

### Solution

so

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

### Analysis of the Solution

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

### Try It 2

If

### Example 3: Determining Inverse Relationships for Power Functions

If

### Solution

No, the functions are not inverses.

### Analysis of the Solution

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

### Try It 3

If

## Finding Domain and Range of Inverse Functions

The outputs of the functionWhen 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

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

If

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

**Q & A**

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

*No. If two supposedly different functions, say, *

*$g$*

*and$h$, both meet the definition of being inverses of another function $f$, then you can prove that $g=h$. 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

The domain of

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

- 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.
- 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 |

$f\left(x\right)=c$ |
$f\left(x\right)=x$ |
$f\left(x\right)={x}^{2}$ |
$f\left(x\right)={x}^{3}$ |
$f\left(x\right)=\frac{1}{x}$ |

Reciprocal squared | Cube root | Square root | Absolute value | |

$f\left(x\right)=\frac{1}{{x}^{2}}$ |
$f\left(x\right)=\sqrt[3]{x}$ |
$f\left(x\right)=\sqrt{x}$ |
$f\left(x\right)=|x|$ |

### 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

The reciprocal-squared function can be restricted to the domain

### 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.### Try It 4

The domain of function

## 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

$t\text{ (minutes)}$ |
30 | 50 | 70 | 90 |

$f\left(t\right)\text{ (miles)}$ |
20 | 40 | 60 | 70 |

### Solution

The inverse function takes an output of

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

### Try It 5

Using the table below, find and interpret (a)

$t\text{ (minutes)}$ |
30 | 50 | 60 | 70 | 90 |

$f\left(t\right)\text{ (miles)}$ |
20 | 40 | 50 | 60 | 70 |

## 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.

- Find the desired input on the
*y*-axis of the given graph. - 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### Solution

To evaluate

*x*-axis and find the corresponding output value on the

*y*-axis. The point

*x*for which

### Try It 6

Using the graph in Example 6, (a) find

## 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,

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

- Make sure $f$is a one-to-one function.
- Solve for $x$.
- Interchange $x$and$y$.

### 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.

### Solution

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

then

In this case, we introduced a function

### Example 8: Solving to Find an Inverse Function

Find the inverse of the function

### Solution

So

### Analysis of the Solution

The domain and range of

$x$ |
1 | 2 | 5 | ${f}^{-1}\left(y\right)$ |

$f\left(x\right)$ |
3 | 2 | 5 | $y$ |

### Example 9: Solving to Find an Inverse with Radicals

Find the inverse of the function

### Solution

So

The domain of

### Analysis of the Solution

The formula we found for

### Try It 8

What is the inverse of the function

**Restricting the domain** to

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

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### 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

**Q & A **

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

*Yes. If *

*$f={f}^{-1}$*

*, then$f\left(f\left(x\right)\right)=x$, and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because*

*Any function *

*$f\left(x\right)=c-x$*

*, where$c$ is a constant, is also equal to its own inverse.*

## Key Concepts

- If $g\left(x\right)$is the inverse of$f\left(x\right)$, then
- $g\left(f\left(x\right)\right)=f\left(g\left(x\right)\right)=x$.
- 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 $y=f\left(x\right)$for$x$as a function of$y$. Then exchange the labels$x$and$y$.
- The graph of an inverse function is the reflection of the graph of the original function across the line $y=x$.

## Glossary

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

## 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

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

For the following exercises, find

7.

8.

9.

10.

11.

12.

For the following exercises, find a domain on which each function

13.

14.

15.

16. Given

a. Find

b. What does the answer tell us about the relationship between

For the following exercises, use function composition to verify that

17.

18.

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

19.

20.

21.

22.

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

23.

24.

For the following exercises, use the graph of

25. Find

26. Solve

27. Find

28. Solve

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

29. Sketch the graph of

30. Find

31. If the complete graph of

32. If the complete graph of

For the following exercises, evaluate or solve, assuming that the function

33. If

34. If

35. If

36. If

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

$x$ |
$f\left(x\right)$ |

0 | 8 |

1 | 0 |

2 | 7 |

3 | 4 |

4 | 2 |

5 | 6 |

6 | 5 |

7 | 3 |

8 | 9 |

9 | 1 |

38. Solve

39. Find

40. Solve

41. Use the tabular representation of

$x$ |
3 | 6 | 9 | 13 | 14 |

$f\left(x\right)$ |
1 | 4 | 7 | 12 | 16 |

42.

43.

44. Find the inverse function of

45. To convert from

46. The circumference

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,

## Analysis of the Solution

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