Composite Functions (OCR A Level Maths A: Pure): Revision Note

Composite functions

What is a composite function?

• A composite function is where one function is applied after another function

• The ‘output’ of one function will be the ‘input’ of the next one

• Sometimes called function-of-a-function

• A composite function can be denoted

  • $fg\left(x\right)$

  • $f\left(g\right(x\left)\right)$

  • $f\left[g\left(x\right)\right]$

  • $(f\circ g)\left(x\right)$

  • All of these mean “$f$ of $g\left(x\right)$” 

How do I use composite functions?

• Recognise the notation

  • fg(x) means “f of g of x”

  The order matters

  • First apply g to x to get $g\left(x\right)$

  • Then apply f to the previous output to get $f\left(g\right(x\left)\right)$

  • Always start with the function closest to the variable

  • $fg\left(x\right)$ is not usually equal to $gf\left(x\right)$

What are special cases of composite functions?

• fg(x) and gf(x) are generally different but can sometimes be the same

• ff(x) is written as f²(x)

• Inverse functions ff^-1(x) = f^-1f(x) = x