COMPOSITION OF FUNCTIONS

 

See this earlier lesson for a thorough introduction to function composition.

Make sure you can do all the exercises there before moving on to this page.

This current lesson 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