Composite Functions

Composite Functions

GCSE(H),

A composite function are two or more functions combined together. The output from one function is used as the input for the other.

If there are two functions f(x) and g(x), and the output from g(x) is to be used as the input for f(x), then the composite function is f(g(x)). It is more normally written simply as fg(x).

Work from the innermost function out; for fg(x), evaluate the function g(x) first, then f(x) using the answer from g(x). Order is important: fg(x) is different to gf(x).

Examples

1. If f(x) = x2 and g(x) = 2x - 4, work out fg(5).

Answer: 36

Work out g(5) first: 2 x 5 - 4 = 6.

This is the input to f(x): f(6) = 62 = 36.

2. Write fg(x), in terms of x, when f(x) = 3x - 4 and g(x) = 4 - 3x.

Answer: 8 - 9x

Substitute the output from g(x) into f(x):

fg(x) = 3(g(x)) - 4

= 3(4 - 3x) - 4

= 12 - 9x - 4

= 8 - 9x