# What is Composition of Functions ?

Here you will learn what is composition of functions with properties and examples.

Let’s begin –

## What is Composition of Functions ?

Let f : A $$\rightarrow$$ B & g : f : B $$\rightarrow$$ C be two functions. Then the function gof : f : A $$\rightarrow$$ C defined by (gof)(x) = g(f(x)) $$\forall$$ x $$\in$$ A is called the composite of the two function f & g.

## Properties :

(a) In general composite of functions is not commutative i.e. gof $$\ne$$ fog.

(b) The composition of functions is associative i.e. if f, g, h are three functions such that fo(goh) & (fog)oh are defined, then fo(goh) = (fog)oh.

(c) The composition of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined, then gof is also a bijection.

Example : If f(x) = $$x^2$$ + 1, g(x) = $$1\over x-1$$, then find (fog)(x).

Solution : Now (fog)(x) = f(g(x)) = f($$1\over x-1$$) = f(z), where z = $$1\over x-1$$

= $$z^2 + 1$$ [ $$\because$$ f(x) = $$x^2 + 1$$ ]

= $$({1\over x-1})^2$$ + 1 = $$1\over {(x-1)^2}$$ + 1