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.
This page adds information and exercises concerning the domain of a composite function.
DEFINITION
function composition
The function
[beautiful math coming... please be patient]$\,g\circ f\,$ (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:

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
$g(x) = x5\,$ 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}\ \ x5\in[0,2]\}\cr
&= \{x\ \ x\in [4,6]\ \ \text{and}\ \ x\in[5,7]\}\cr
&= [5,6]
\end{align}
$
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
