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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)
    xf-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

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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-1(x). It relates an input 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-1(x) shown in the note above, (with the first swapped around so the input is on the left):



  • 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).