# How to Find the Inverse of a Function Using Algebra (Mathematics Lesson)

# How to Find the Inverse of a Function Using Algebra

The inverse of a function can be found by rearranging the function using algebra and then relabelling the input and output.# A Real Example of How to Find the Inverse of a Function Using Algebra

**Question:**What is the inverse function of the function:

**Step 1:**Rearrange the function so the input

**x**is the subject of the equation.

Initially, the output is written in terms of the input,

**f(x) =**

Rearrange the equation using algebra (in this case algebra with addition).

The input is now written in terms of the output,

**x =**

**Step 2:**Relabel the function so the input becomes the output and the output becomes the input.

- The old input
**x**becomes our new output**f**^{-1}(x)

**x**→**f**^{-1}(x) - The old output
**f(x)**becomes our new input**x**.

**f(x)**→**x**

The inverse function is

# Another Real Example of How to Find the Inverse of a Function Using Algebra

The slider below shows a real example of how to find the inverse of a function using algebra.##### Interactive Test

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##### Note

**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**. It relates an input

^{-1}(x)**x**to an output

**f**:

^{-1}(x)**WHY DO WE RELABEL THE INPUT AND OUTPUT?**

In

**Step 2**(see left), we relabel the input as the output and the output as the input. Why is this?

Consider the two arrow diagrams of

**f**shown in the note above, (with the first swapped around so the input is on the left):

^{-1}(x)- The
**output**of the*function***f(x)**is equivalent to the**input**of the*inverse function***x**. - The
**input**of the*function***x**is equivalent to the**output**of the*inverse function***f**.^{-1}(x)