# What Is an Inverse Function? (Mathematics Lesson)

# What Is an Inverse Function?

An inverse function is itself a function which reverses a function.If a function

**f(x)**maps an input

**x**to an output

**f(x)**...

... an inverse function takes the output

**f(x)**back to the input

**x**:

An inverse function is denoted

**f**.

^{-1}(x)# Visualizing Inverse Functions

It is useful to visualize functions as machines, which takes in an input, x, and processes it into an output, f(x).An inverse function runs this machine backwards, taking the output back to the input:

# How to Find the Inverse of a Function

There are 3 ways to find the inverse of a function.**Find the inverse function using algebra.**

The inverse of a function can be found by rearranging the function using algebra and then relabelling the input and output.

Read more about how to find the inverse of a function using algebra**Find the inverse function using a number machine.**

If a function is represented by a number machine, the inverse function is found by reversing the machine, and changing all its operations to their opposites.

Read more about how to find the inverse of a function using a number machine**Find the inverse function using a graph.**

If a function is plotted on a graph, the inverse function can be found by reflecting the function in the line**y = x**.

Read more about how to find the inverse of a function using a graph

# More on Inverse Functions

The slider below gives some more information about inverse functions.##### Interactive Test

**show**

##### Note

# AN EXAMPLE OF AN INVERSE FUNCTION

A function takes an input and adds 1 to it to produce an output.If it takes in an input

**x = 1**the output is

**f(x) = 2**:

which can be written:

The inverse of this function will be:

By inputting a

**2**, we get the original input

**1**as as an output:

which can be written:

# INVERSE FUNCTIONS ARE THEMSELVES FUNCTIONS

Don't be fooled into thinking that inverse functions are a completely different thing from functions.An inverse function is itself a function.

Inverse functions have the same properties as functions and behave just like functions.

An inverse function is only an inverse relative to the function it is the inverse of.

A person is a husband relative to their wife, but they are still a person.

**TO 'INVERT' A FUNCTION**

The verb which expresses 'to find the inverse of a function' is 'to invert'.

# INVERTING AN INVERSE FUNCTION

Take an function**f(x)**and invert it to

**f**:

^{-1}(x)Now invert the inverse function:

We are back at the original function,

**f(x)**.

If you invert a function twice, you get back to the original function.

**WHAT IS A FUNCTION?**

A function is a relation between an input and an output.

The input is often denoted

**x**and the function

**f(x)**.

Consider the function below:

- Let the input be denoted by the variable
**x**. - The relation the input plus one:
**x + 1**. - The output is the result of adding one to the input.