Inverse Functions (OCR GCSE Maths: Higher): 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

• The inverse of $\mathrm{input}\to \boxed{\times 6}\to \boxed{+4}\to \mathrm{output}$ is $\mathrm{input}\to \boxed{-4}\to \boxed{÷6}\to \mathrm{output}$

  • The order of the boxes is reversed

  • The operation is replaced by its inverse operation

    • × ↔ ÷ and  + ↔ -

• If a number goes through a function, then that result goes through the inverse function, you get back the same number again!

How do I find an inverse function algebraically?

• If putting x into a function gives out y, then putting y into the inverse function gives back x

• Inverse functions are related to rearranging formulae

  • Let's say a function is $x\to \boxed{\times 2}\to \boxed{+3}\to y$

    • It's inverse is therefore $x\to \boxed{-3}\to \boxed{÷2}\to y$

  • In algebra, the original function is $2x+3=y$

  • Look what happens when you make x the subject

    • $x=\frac{y-3}{2}$

    • This is the function $y\to \boxed{-3}\to \boxed{÷2}\to x$ 

    • The operations are identical to that of the inverse function! (the input here is a y though)

  • Swapping the x and y shows the inverse function more clearly

    • $x\to \boxed{-3}\to \boxed{÷2}\to y$