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.
Also Read : Types of Functions in Maths – Domain and Range
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
