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

The Lesson

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.

Question

Find the inverse function of the function below.

Step-by-Step:

1

Rearrange the function to find "x =".
f(x) = ½x + 1
f(x) − 1 = ½x + 1 − 1 Subtract 1 from both sides
f(x) − 1 = ½x
2 × ( f(x) − 1 ) = 2 × ½x Multiply both sides by 2
2( f(x) − 1 ) = x
x = 2( f(x) − 1 )
We have rearranged f(x) = ½x + 1 to find what "x = ". x = 2( f(x) − 1 )

2

Replace x with f−1(x).
f−1(x) = 2( f(x) − 1 )

3

Replace f(x) with x.
f−1(x) = 2( x − 1 )

Answer:

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

Lesson Slides

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Why Do We Relabel the Input and the Output?

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).

If we compare these two conceptions...

...we see that we need to replace f(x) with x and x with f−1(x).
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See Also

What is a function? What are Cartesian coordinates?