Learning Objectives
- Find the domain of a composite function
- Decompose a composite function
As we discussed previously, the domain of a composite function such as [latex]f\circ g[/latex] is dependent on the domain of [latex]g[/latex] and the domain of [latex]f[/latex]. It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as [latex]f\circ g[/latex]. Let us assume we know the domains of the functions [latex]f[/latex] and [latex]g[/latex] separately. If we write the composite function for an input [latex]x[/latex] as [latex]f\left(g\left(x\right)\right)[/latex], we can see right away that [latex]x[/latex] must be a member of the domain of [latex]g[/latex] in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that [latex]g\left(x\right)[/latex] must be a member of the domain of [latex]f[/latex], otherwise the second function evaluation in [latex]f\left(g\left(x\right)\right)[/latex] cannot be completed, and the expression is still undefined. Thus the domain of [latex]f\circ g[/latex] consists of only those inputs in the domain of [latex]g[/latex] that produce outputs from [latex]g[/latex] belonging to the domain of [latex]f[/latex]. Note that the domain of [latex]f[/latex] composed with [latex]g[/latex] is the set of all [latex]x[/latex] such that [latex]x[/latex] is in the domain of [latex]g[/latex] and [latex]g\left(x\right)[/latex] is in the domain of [latex]f[/latex].
A General Note: Domain of a Composite Function
The domain of a composite function [latex]f\left(g\left(x\right)\right)[/latex] is the set of those inputs [latex]x[/latex] in the domain of [latex]g[/latex] for which [latex]g\left(x\right)[/latex] is in the domain of [latex]f[/latex].
How To: Given a function composition [latex]f\left(g\left(x\right)\right)[/latex], determine its domain.
- Find the domain of g.
- Find the domain of f.
- Find those inputs, x, in the domain of g for which g(x) is in the domain of f. That is, exclude those inputs, x, from the domain of g for which g(x) is not in the domain of f. The resulting set is the domain of [latex]f\circ g[/latex].
Example: Finding the Domain of a Composite Function
Find the domain of
[latex]\left(f\circ g\right)\left(x\right)\text{ where}f\left(x\right)=\frac{5}{x - 1}\text{ and }g\left(x\right)=\frac{4}{3x - 2}[/latex]
Example: Finding the Domain of a Composite Function Involving Radicals
Find the domain of
[latex]\left(f\circ g\right)\left(x\right)\text{ where}f\left(x\right)=\sqrt{x+2}\text{ and }g\left(x\right)=\sqrt{3-x}[/latex]
Try It
Find the domain of
[latex]\left(f\circ g\right)\left(x\right)\text{ where}f\left(x\right)=\frac{1}{x - 2}\text{ and }g\left(x\right)=\sqrt{x+4}[/latex]
Try it now
We can use graphs to visualize the domain that results from a composition of two functions.
Graph the two functions below with Desmos.
- [latex]f(x)=\sqrt{3-x}[/latex]
- [latex]g(t) = \sqrt{x+4}[/latex]
Next, create a new function, [latex]h(x) = g(f(x))[/latex]. Based on the graph, what is the domain of this function? Explain why [latex]g(f(x))[/latex] and [latex]f(x)[/latex] have the same domain.
Now define another composition, [latex]p(x) = f(g(x)[/latex]. What is the domain of this function? Explain why you can evaluate [latex]g(10)[/latex], but not [latex]p(10)[/latex].
Decompose a Composite Function
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.
Example: Decomposing a Function
Write [latex]f\left(x\right)=\sqrt{5-{x}^{2}}[/latex] as the composition of two functions.
Try It
Write [latex]f\left(x\right)=\frac{4}{3-\sqrt{4+{x}^{2}}}[/latex] as the composition of two functions.