Composite Functions (DP IB Analysis & Approaches (AA)): Revision Note

Composite functions

What is a composite function?

• A composite function is where a function is applied to another function

  • written as any out of

    • 

    • 

    • 

• The order matters

  • means:

    • First apply to to get 

    • Then apply to the previous output to get 

  • i.e. start with the function closest to

• is not usually equal to 

How do I find the domain of a composite function?

• This is best shown through an example

  • e.g. let and let 

• What is the domain of ?

  • In , an input goes into first, which accepts the inputs

    • i.e. the domain so far is

  • Substitute the inputs into to get the outputs

  • The outputs become the new inputs for

  • Substitute into

    • Not possible: only accepts inputs between and

  • It would only work if

    • to make it fit between and

  • but means the initial domain should have been

  • so is the domain of

How do I find the range of a composite function?

• Using the same example from above, to find the range of

  • substitute the domain (found above) into the composite function (in order)

  • The domain is

    • First substitute into to get

    • Then substitute into to get

    • This means the range of is