Composite Functions (Edexcel IAL Maths)

Spec point — 1.2

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

  • $fg(x)$
  • $f(g(x))$
  • $f [ g ( x ) ]$
  • $(f\circ g)(x)$
  • All of these mean “$f$ of $g(x)$”

How do I use composite functions?

• Recognise the notation The order matters

  • fg(x) means “f of g of x”

  • First apply g to x to get $g(x)$

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

  • Always start with the function closest to the variable

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

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

Exam Hint

• Domain and range are important.In fg(x), the ‘output’ (range) of g must be in the domain of f(x), so fg(x) could exist, but gf(x) may not (or not for some values of x).

Worked Example