Linear and Quadratic Functions: Functions and Relations

Composition of Functions

The last topic that needs to be discussed in terms of functions and relations is the composition of functions. Let f and g be functions of x (f(x) and g(x)). The composition of f and g is written as f of g and is defined as f(g(x)).

One important note. Make sure not to get multiplication confused with composition. This is a commonly made mistake. Composition is denoted with the symbol small open circle while multiplication is denoted with the symbol multiplication dot.

The function f of g is called the composite function. Let’s see how this works.

We are going to work with our two previous functions, f(x) = 2x + 6 and g(x) = 4x – 9.

  1. Find f of g.
    f of g = f(g(x)) = 2(4x – 9) + 6
    g(x) is inserted into the x-value of f(x).
    f of g = 8x – 18 + 6
    f of g = 8x – 12
  2. Find g of f.
    g of f = 4(2x + 6) – 9
    f(x) is inserted into the x-value of g(x).
    g of f = 8x + 24 – 9
    g of f = 8x + 15