Inverse Functions (AQA GCSE Maths): Revision Note

Inverse Functions

What is an inverse function?

• An inverse function does the opposite (reverse) operation of the function it came from

  • E.g. If a function “doubles the number then adds 1”

  • Then its inverse function “subtracts 1, then halves the result”

    • The same inverse operations are used when solving an equation or rearranging a formula

• An inverse function performs the inverse operations in the reverse order

What notation is used for inverse functions?

• The inverse function of is written as

  • For example, if 

  • The inverse function is   or

• If then

  • For example

    • (inputting 3 into gives 7)

    • (inputting 7 into gives back 3)

How do I find an inverse function algebraically?

• The process for finding an inverse function is as follows:

  • Write the function as

    • E.g. The function becomes

  • Swap the s and s to get

    • E.g.

    • The letters change but no terms move

  • Rearrange the expression to make the subject again

    • E.g. becomes so

  • Replace with (or )

    • E.g.

    • This is the inverse function

    • should not appear in the final answer

How are inverse functions and composite functions related?

• The composite function of followed by (or the other way round) cancels out

  • 

    • If you apply a function to x, then apply its inverse function, you get back x

    • Whatever happened to x gets undone

    • f and f^-1 cancel each other out when applied together

• For example, solve  where 

  • Finding the inverse function algebraically in this case is tricky

    • (It is impossible if you haven't studied logarithms!)

  • Instead, you can take of both sides of and use the fact that  cancel each other out:
    

    • which cancels to giving