Composition of Functions of Set

Two functions f: A → B and g: B → C can be composed to give a composition g o f. This is a function from A to C defined by (g o f)(x) = g(f(x))

Example

Let f(x) = x + 2 and g(x) = 2x + 1, find (f o g)(x) and (g o f)(x).

Solution

(f o g)(x) = f(g(x)) = f(2x + 1) = 2x + 1 + 2 = 2x + 3

(g o f)(x) = g (f(x)) = g(x + 2) = 2 (x+2) + 1 = 2x + 5

Hence, (f o g)(x) ≠ (g o f)(x)

Some Facts about Composition

• If f and g are one-to-one then the function (g o f) is also one-to-one.

• If f and g are onto then the function (g o f) is also onto.

• Composition always holds associative property but does not hold commutative property.