# Composition of Functions

The composition of two functions f and g is the new function h, where h(x) = f(g(x)), for all x in the domain of g such that g(x) is in the domain of f. The notation for function composition is h = f • g or h(x) = (f • g)(x) and is read as 'f of g of x'. The procedure is called composition because the new function is composed of the two given functions f and g, where one function is substituted into the other.

## Finding the Composition

Although composition of functions is best illustrated with an example, let us summarize the key steps:

• rewrite f • g as f(g(x));
• replace g(x) with the function that it represents;
• evaluate f by replacing every x with the function that g(x) represents; and
• finally, if given a numerical value of x, evaluate the new function at this value by replacing all remaining x with the given value.

Note: Often f • g ≠ g • f and the two will have different domains. Also, be aware that you can take the composition  of more than two functions: e.g., f(g(k(x))).

Example: Given the function f(x) = x2 and g(x) = x + 3, find f(g(1)) and g(f(1)).

Solution:

f(g(1)) = f(x + 3)                               g(f(x)) = g(x2)

= (x +3)2                                                     = x2 + 3

f(g(1)) = (1 + 3)2                              g(f(1)) = 12 + 3

= 16                                                 = 4

Notice that  f • g ≠ g • f

Composition of Function Example 1:

Composition of Function Example 2:

Composition of Function Example 3:

Composition of Function Example 4:

Composition of Function Example 5:

Composition of Function Example 6: