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: