Composite Functions (Cambridge (CIE) AS Maths) : 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

  • 

  • 

  • 

  • 

  • All of these mean “ of ” 

How do I work with composite functions?

• Recognise the notation

  • fg(x) means “f of g of x”

  The order matters

  • First apply g to x to get 

  • Then apply f to the previous output to get 

  • Always start with the function closest to the variable

  • is not usually equal to 

Special cases

• 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