# COMPOSITION OF FUNCTIONS

• PRACTICE (online exercises and printable worksheets)

See this earlier web exercise for a thorough introduction to function composition.
Make sure you can do all the exercises there before moving on to this page.

 DEFINITION function composition The function $\,g\circ f\,$ (read as ‘$\,g\,$ circle $\,f\$’) is defined by: $$(g\circ f)(x) := g(f(x))$$ the function $\,g\circ f\$: $\,f\,$ is ‘closest to’ the input, and acts first; $\,g\,$ acts second The domain of $\,g\circ f\,$ is the set of inputs $\,x\,$ with two properties: $\,f\,$ must know how to act on $\,x\,$; that is, $\,x\in\text{dom}(f)\,$ $\,g\,$ must know how to act on $\,f(x)\,$; that is, $\,f(x)\in\text{dom}(g)\,$ Thus: $$\text{dom}(g\circ f)= \{x\ |\ x\in\text{dom}(f)\ \ \text{and}\ \ f(x)\in\text{dom}(g)\}$$

## EXAMPLE

This example is contrived to give practice with the domain of a composite function.

PROBLEM:
Suppose that $\,f\,$ is the ‘add $\,3\,$’ function with a restricted domain:   $\text{dom}(f) = [0,2]\$.
Thus, $\,f\,$ only knows how to act on the numbers $\,0\le x\le 2\,$.

Suppose that $\,g\,$ is the ‘subtract $\,5\,$’ function with a restricted domain:   $\text{dom}(g) = [4,6]\,$.

Find the domains of both $\,g\circ f\,$ and $\,f\circ g\,$.

SOLUTION:
We have:
$f(x) = x+3$ with $\text{dom}(f) = [0,2]$
and
$g(x) = x-5\,$ with $\text{dom}(g) = [4,6]$

Firstly, observe that the following sentences are equivalent:

$\displaystyle \begin{gather} f(x)\in \text{dom}(g)\cr x+3\in [4,6]\cr 4\le x+3 \le 6\cr 1\le x\le 3\cr x\in [1,3] \end{gather}$

Then,

\displaystyle \begin{align} \text{dom}(g\circ f) &= \{x\ |\ x\in\text{dom}(f)\ \ \text{and}\ \ f(x)\in\text{dom}(g)\}\cr &= \{x\ |\ x\in [0,2]\ \ \text{and}\ \ x\in[1,3]\}\cr &= \{x\ |\ x\in ([0,2]\cap[1,3])\}\cr &= \{x\ |\ x\in [1,2]\}\cr &= [1,2] \end{align}

Similarly (and a bit more compactly),

\displaystyle \begin{align} \text{dom}(f\circ g) &= \{x\ |\ x\in\text{dom}(g)\ \ \text{and}\ \ g(x)\in\text{dom}(f)\}\cr &= \{x\ |\ x\in [4,6]\ \ \text{and}\ \ x-5\in[0,2]\}\cr &= \{x\ |\ x\in [4,6]\ \ \text{and}\ \ x\in[5,7]\}\cr &= [5,6] \end{align}

Master the ideas from this section