how to find inverse of a function

Learn How to Find the Inverse of a Function Using 3 Easy Steps

The following step-by-step guide will show you how to find the inverse of any function! (Algebra)

Welcome to this free lesson guide that accompanies this Finding the Inverse of a Function Tutorial where you will learn the answers to the following key questions and information:

What is the inverse of a function?
What does the graph of the inverse of a function look like?
How can I find the inverse of a function algebraically?
How can I find the inverse of a function graphically?

This Complete Guide to Finding the Inverse of a Function includes several examples, a step-by-step tutorial and an animated video tutorial.

Intro to Finding the Inverse of a Function

Before you work on a find the inverse of a function examples, let's quickly review some important information:

Notation: The following notation is used to denote a function (left) and it's inverse (right). Note that the -1 use to denote an inverse function is not an exponent.

Notice that the terms are in reverse order!

What is a function?

By definition, a function is a relation that maps X onto Y.

And what is the inverse of a function?

An inverse function is a relation that maps Y onto X.

Notice the switch?

Figure 1: The parent function for absolute value.

You can think of the relationship of a function and it's inverse as a situation where the x and y values reverse positions.

For example, let's take a look at the graph of the function f(x)=x^3 and it's inverse.

Take a look at the table of the original function and it's inverse. Notice how the x and y columns have reversed!

Definition: The inverse of a function is it's reflection over the line y=x.

Keep this relationship in mind as we look at an example of how to find the inverse of a function algebraically.

Finding the Inverse of a Function Example

We will be using the following 3-step process that can be used to find the inverse of any function:

STEP ONE: Rewrite f(x)= as y=

If the function that you want to find the inverse of is not already expressed in y= form, simply replace f(x)= with y= as follows (since f(x) and y both mean the same thing: the output of the function):

Set up two equations (positive and negative) and ditch the absolute value bars.

STEP ONE: Swap X and Y

Now that you have the function in y= form, the next step is to rewrite a new function using the old function where you swap the positions of x and y as follows:

This new function with the swapped X and Y positions is the inverse function, but there's still one more step!

STEP THREE: Solve for y (get it by itself!)

The final step is to rearrange the function to isolate y (get it by itself) using algebra as follows:

Left: replace x with 3. Right: replace x with -9

It's ok the leave the left side as (x+4)/7

Once you have y= by itself, you have found the inverse of the function!

Final Answer: The inverse of f(x)=7x-4 is f^-1(x)=(x+4)/7

Graphs of Inverse Functions

Remember earlier when we said the inverse function graph is the graph of the original function reflected over the line y=x? Let's take a further look at what that means using the last example:

Below, Figure 1 represents the graph of the original function y=7x-4 and Figure 2 is the graph of the inverse y=(x+4)/7

Now let's take a look at both lines on the same graph. Note that the original function is blue and the inverse is red this time (Figure 3) and then add the line y=x to the same graph (Figure 4).

Can you see the reflection over the line y=x?

This relationship applies to any function and it's inverse and it should help you to understand why the 3-step process that you used earlier works for finding the inverse of any function!