COMPOSITION OF FUNCTIONS

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:
  • $\,f\,$ 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
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
writing a function as a composition
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3
AVAILABLE MASTERED IN PROGRESS