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Given a function

sometimes it is possible to find another function that undoes

This particular function is called the inverse function of

This lesson will discuss how to verify whether two functions are inverse and will explain the procedure to find the inverse of a linear function.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Equation
Solving Multi-Step Equations
Literal Equation
Linear Function
Slope-Intercept Form
Graphing Linear Functions in Slope-Intercept Form
Function Notation

Challenge

Undoing a Function

Consider a function

that takes any real number and returns a number greater by two units.

What would be a function that undoes

Discussion

Inverse Relation

An inverse relation is a relation that interchanges the input and output values of a specific relation. In other words, it is the relation that undoes a certain relation. Consider the mapping diagram that shows a relation

that takes elements from set

and returns elements from set

The relation that undoes

In this case,

takes elements from set

and returns elements from set

Relations can also be represented by a collection of ordered pairs. Here, the inverse of a relation is obtained by interchanging the elements of the ordered pairs of the original relation.

Graphically, the inverse of a relation is a reflection across the line

If the inverse relation of a function is also a function, then it is called an inverse function.

Discussion

Inverse of a Function

Every function has an inverse relation. If this inverse relation is also a function, then it is called an inverse function. In other words, the inverse of a function is another function such that they undo each other.

Therefore, and are inverses of each other. Also, if is the input of a function and its corresponding output, then is the input of and its corresponding output.

Consider a function

and its inverse

Why

Showing That the Functions Are Inverse.

The functions

and

will be shown to undo each other. To do so, it needs to be proven that

and that

To start, the first equality will be proven. First, the definition of

will be used.

Now, in the above equation,

will be substituted for

Substitute

▼

Simplify left-hand side

DenomMultFracToNumber

RemovePar

Remove parentheses

SubTerm

Subtract term

$x=x\ {\color{#009600}{\bm{\Large{\checkmark }}}}$

A similar procedure can be performed to show that

	Definition of First Function	Substitute Second Function	Simplify
			$x=x\ {\color{#009600}{\bm{\Large{\checkmark }}}}$
			$x=x\ {\color{#009600}{\bm{\Large{\checkmark }}}}$

Therefore, and undo each other. The graphs of these functions are each other's reflection across the line This means that the points on the graph of are the reversed points on the graph of

In general, the inverse of a linear function is also a linear function.

Example

Using Graphs to Determine if Two Functions Are Inverse Functions

Kriz enjoys playing video games with their friends. For their birthday they received a copy of Mathleaks: The Adventure, a math-based video game.

Kriz is very close to beating this level. They need to graph the functions given below and determine whether they are inverse functions.

Help Kriz beat the level and get one step closer beating the game by mastering math!

Answer

Graph:

Are $\bm{f}$ and $\bm{g}$ Inverse Functions? Yes.

Hint

Graph both functions on the same coordinate plane and see if they are each other's reflection across the line

Solution

Since and are linear functions written in slope-intercept form, they can both be graphed using their slope and intercept.

Now, it can be seen whether the lines are each other's reflection across

The lines are indeed each other's reflection across

Therefore,

and

are inverse functions.

Pop Quiz

Determining if Two Functions Are Inverses by Looking at Their Graphs

It was discussed previously that if the graphs of two functions are each other's reflection across then they are inverse functions. Therefore, it can be determined if two functions are inverse just by looking at their graphs. Determine whether the following linear functions are inverse by looking at the lines.

Do the lines shown correspond to inverse functions?

Example

Verifying Inverse Functions by Composition

If Paulina gets an A on a math test, her mother will buy her a new saxophone.

To get an A, Paulina must answer two questions correctly. Help her get the new saxophone!

a Given

and

calculate

and

and determine whether

and

are inverse functions.

Are

and

inverse functions?

b Given

and

calculate

and

and determine whether

and

are inverse functions.

Are

and

inverse functions?

Hint

a If

and

then

and

are inverse functions.

b If

and

then

and

are inverse functions.

Solution

a Consider the given functions.

The composite function

will be found first. To do so, the definition of

will be used.

Next,

can be substituted in the above equation.

Substitute

▼

Simplify right-hand side

Distr

Distribute

AssociativePropMult

Associative Property of Multiplication

DenomMultFracToNumber

Multiply

AddTerms

Add terms

By following the same procedure, an expression for

can be found.

	Definition of First Function	Substitute Second Function	Simplify

It has been found that both and are equal to Therefore, and are inverse functions.

b Consider the given functions.

By following the same procedure as in Part A, the expressions for

and

will be found.

	Definition of First Function	Substitute Second Function	Simplify

It has been found that neither nor is equal to Therefore, and are not inverse functions.

Pop Quiz

Determining if Two Functions Are Inverse by Composition

For the functions and use a composition to determine whether they are inverse functions.

determine if two functions are inverse by compounding

Discussion

Finding the Inverse of a Function Algebraically

A function can be represented by a table of values, a graph, a mapping diagram, or a function rule, among other ways. Depending on how the function is presented, finding its inverse can be done in different ways. When a function rule is given, finding the inverse algebraically is advantageous. Consider the following example function.

There is a series of steps to follow in order to find the inverse function

Replace

With

To begin, since

describes the input-output relationship of the function, replace

with

in the function rule.

Switch

and

Because the inverse of a function reverses

and

the variables can be switched. Notice that every other piece in the function rule remains the same.

Solve for

Solve the resulting equation from the previous step for

This will involve using the inverse operations.

▼

Solve for

Rearrange equation

Replace

With

Just as

shows the input-output relationship of

so does

Therefore, replacing

with

gives the rule for the inverse of

Notice that in

the input is multiplied by

decreased by

and divided by

From the rule of

it can be seen that

undergoes the inverse of these operations in the reverse order. Specifically,

is multiplied by

increased by

and divided by

Example

Finding the Inverse of a Function

Zosia lives in Honolulu, Hawaii.

She is obsessed with math. While watching a surfer catch some gnarly waves, she came up with a function that models the distance of the surfer from the shore.

Here,

is the distance of the surfer from the shore after

seconds of riding a wave. Given the distance from shore, Zosia now wants to finds the function that gives the time, in seconds, the surfer has been riding a wave. To do so, she needs to find

the inverse of

Help Zosia find this function! Write the answer in slope-intercept form.

Hint

Start by replacing with After that, switch the and variables. Then, solve the obtained equation for Finally, replace with

Solution

To find the inverse of the function, there are four steps to follow.

Replace with
Switch and
Solve the obtained equation for
Replace with

These steps will be done one at a time.

Step

Here, the variable

will be substituted for

in the given function.

Step

In this step, the variables

and

must be switched.

Step

Here, the equation obtained in the previous step will be solved for

▼

Solve for

Distribute

Commutative Property of Multiplication

Calculate quotient

Rearrange equation

Step

Finally,

will be substituted for the

variable in the equation obtained in the previous step.

Pop Quiz

Finding the Inverse Function

Find the inverse of the given function. Write the answer in slope-intercept form — in the format If the slope and intercept are not integers, express them as decimal numbers. If necessary, round them to two decimal places.

finding the inverse of a linear function

Closure

Undoing a Function

In this lesson, it has been seen that the function that undoes a function

is its inverse

Consider the function given in the challenge presented at the beginning of the lesson.

Find the inverse function of

Hint

Start by replacing with After that, switch the and variables. Then, solve the obtained equation for and finally replace with

Solution

To find the inverse of the function, there are four steps to follow.

Replace with
Switch and
Solve the obtained equation for
Replace with

These steps will be done one at a time.

Step

Here, the variable

will be substituted for

in the given function.

Step

In this step, the variables

and

must be switched.

Step

Here, the equation obtained in the previous step will be solved for

Step

Finally,

will be substituted for the

variable in the equation obtained in the previous step.

It has been found that the inverse function of

and therefore the function that undoes it, is

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Catch-Up and Review

Why

Answer

Hint

Solution

Hint

Solution

Hint

Solution

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Step

Step

Step

Hint

Solution

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