# Composite Functions

## Composite Functions:

Consider three non-void sets A, B, and C and let f be a function from A to B and g be a function from B to C. We can combine these two functions to get a new function which may be called the composite or the resultant of these two functions.

The usual way of composition is to first apply f to x to get f(x) and then apply g to f(x) to get the resultant image. Thus, if f: A → B and g: B → C be the two functions, then the composite of f and g, denoted by gof is the function gof: A → C defined by

The composite of two functions is sometimes known as the function of a function.

The composite function gof is defined only if, for each x ∈ A, f(x) becomes an element of g so that the g-image of f(x) may be obtained. Thus, for the composition gof to exist, the range of f must be a subset of the domain of g. Similarly, fog exists if the range of g is a subset of the domain of f.

Some Properties of Composite Functions:

• The composition of functions is not commutative i.e., fog ≠ gof.
• The composition of functions is associative i.e., if f, g and h are three functions such that the composition (fog)oh and fo(goh) exist, then [(fog)oh] (x) = [fo(goh)] (x).
• The composition of any function with the identity function is the function itself, i.e., if IA: A → A and f: A → B, then foIA = f. Now, if IB: B → B and f: A → B, then foIB = f.
• The composition of two bijections is a bijection i.e., if f and g are two bijections then fog, as well as gof, are also bijections.