Composite Functions (CIE A Level Maths: Pure 1)

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