Finding the Inverse of a Function
(KS4, Year 10)
The LessonAn 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 FunctionFinding the inverse of a function is easy.
QuestionFind the inverse function of the function below.
Rearrange the function to find "x =".
We have rearranged f(x) = ½x + 1 to find what "x = ".
x = 2( f(x) − 1 )
|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 )|
Replace x with f−1(x).
f−1(x) = 2( f(x) − 1 )
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 SlidesThe slider below explains more about how to find the inverse of a function. Open the slider in a new tab
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).
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).
...we see that we need to replace f(x) with x and x with f−1(x).
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