Introduction - Composite functions

A composite function is a function obtained by applying functions one after another.

Consider the functions $f$ and $g$ defined by

\begin{align*} &f: x \mapsto x+1, \quad &&\textrm{for } x \in \mathbb{R}, \\ &g: x \mapsto x^2, \quad &&\textrm{for } x \in \mathbb{R}. \end{align*}

Then we have $fg(3)=f(3^2)=9+1=10$ .

The composite function $fg$ represents the function obtained by first applying $g$ followed by $f$ .

On the other hand $gf(3)=g(3+1)=4^2=16$ so the composite function $gf$ represents first applying $f$ followed by $g$ .

\begin{align*} &f: x \mapsto x+1, \quad &&\textrm{for } x \in \mathbb{R}, \\ &g: x \mapsto x^2, \quad &&\textrm{for } x \in \mathbb{R}. \end{align*}