Composite Functions (Cambridge (CIE) IGCSE International Maths)

Composite Functions

What is a composite function?

• A composite function is a function applied to the output of another function

  • The input goes through the 1st function to become an output

  • This output goes through the 2nd function to become a new output

What notation is used for composite functions?

• If and are two functions, then

  • is a composite function

  • It means the input goes through function first

    • This gives the output

    • Then this output, , becomes the input of function , giving

  • is the shorthand notation used for

    • It means do first, then

    • The order of applying the functions goes from right to left

    • (the letter nearest the bracket goes first)

    • This is often the opposite of what people expect!

  • means do first then second

  • means apply twice!

    • This can be written

    • This does not mean the same as

How do I substitute numbers into composite functions?

• If you are putting a number into a composite function

  • put the number into the function closest to (x)

  • then make the output of the first function the input of the second function

• For example, if and

  • to find :

    • Put the 2 in as the input of first

    • 

    • Then put 5 in as the input of

    • So

  • to find :

    • Put the 2 in as the input of first

    • 

    • Then put in as the input of

    • So

  • to find :

    • 

    • 

    • so

How do I find composite functions algebraically?

• If you are using algebra, substitute the whole algebraic expression as your input

  • For example, if  and 

    • 

    • 

    • which simplifies to