Finding the Inverse of a Function
(KS4, Year 10)

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An inverse function is a function that reverses another function. If we have a function f(x), we can find the inverse function f⁻¹(x).

How to Find the Inverse of a Function

Finding the inverse of a function is easy.

Find the inverse function of the function below. find inverse function example

Rearrange the function to find "x =".

We have rearranged f(x) = ½x + 1 to find what "x = ". x = 2( f(x) − 1 )

Replace x with f⁻¹(x).

f⁻¹(x) = 2( f(x) − 1 )

Replace f(x) with x.

f⁻¹(x) = 2( x − 1 )

The inverse of the function f(x) = ½x + 1 is: f⁻¹(x) = 2(x − 1).

The slider below explains more about how to find the inverse of a function.

When we find the inverse of a function, we replace:

the input of the function (x) with the output of the inverse function (f⁻¹(x)), and
the output of the function (f(x)) with the input of the inverse function (x).

This comes from two conceptions of an inverse function.

An inverse function reverses a function, relating the function's output f(x) to its input x. We can think of f(x) being the input to the inverse function and x being its output.
An inverse function is a function. Using functional notation, an inverse function relates an input x to an output f⁻¹(x).