Composite Functions (Cambridge O Level Additional Maths)

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
  •  means “f of g of x”
  The order matters
  
  • First apply  to  to get 
  • Then apply to  to get
  • Always start with the function closest to the variable
  • is not usually equal to 

Special cases

• and  are generally different but can sometimes be the same
•  is written as
  • Note that trig functions are exceptions to this rule
    • e.g.  means  not 
• For inverse functions, 

How do I find the domain and range of composite functions?

• Use logic to determine the domain and range of a composite function
• For  the first function to be applied will be 
  • So, at best, the domain of  will be the same as the domain of 
• However, for this to be the case, the range of  must be contained within the domain of
  • If this is not the case, then restrictions on the domain of  will be required
• Similarly, at best, the range of  will be the same as the range of 
  • But if the domain of  has been affected, the range of  will also be affected