71 Find the domain of 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.

  1. Find the domain of g.
  2. Find the domain of f.
  3. 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 8: 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]

Solution

The domain of [latex]g\left(x\right)\\[/latex] consists of all real numbers except [latex]x=\frac{2}{3}\\[/latex], since that input value would cause us to divide by 0. Likewise, the domain of [latex]f[/latex] consists of all real numbers except 1. So we need to exclude from the domain of [latex]g\left(x\right)\\[/latex] that value of [latex]x[/latex] for which [latex]g\left(x\right)=1\\[/latex].

[latex]\begin{cases}\frac{4}{3x - 2}=1\hfill \\ 4=3x - 2\hfill \\ 6=3x\hfill \\ x=2\hfill \end{cases}\\[/latex]

So the domain of [latex]f\circ g[/latex] is the set of all real numbers except [latex]\frac{2}{3}\\[/latex] and [latex]2[/latex]. This means that

[latex]x\ne \frac{2}{3}\text{or}x\ne 2\\[/latex]

We can write this in interval notation as

[latex]\left(-\infty ,\frac{2}{3}\right)\cup \left(\frac{2}{3},2\right)\cup \left(2,\infty \right)\\[/latex]

Example 9: 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]

Solution

Because we cannot take the square root of a negative number, the domain of [latex]g[/latex] is [latex]\left(-\infty ,3\right]\\[/latex]. Now we check the domain of the composite function

[latex]\left(f\circ g\right)\left(x\right)=\sqrt{3-x+2}\text{ or}\left(f\circ g\right)\left(x\right)=\sqrt{5-x}\\[/latex]

The domain of this function is [latex]\left(-\infty ,5\right]\\[/latex]. To find the domain of [latex]f\circ g\\[/latex], we ask ourselves if there are any further restrictions offered by the domain of the composite function. The answer is no, since [latex]\left(-\infty ,3\right]\\[/latex] is a proper subset of the domain of [latex]f\circ g\\[/latex]. This means the domain of [latex]f\circ g\\[/latex] is the same as the domain of [latex]g[/latex], namely, [latex]\left(-\infty ,3\right]\\[/latex].

Analysis of the Solution

This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in finding the domain of a composite function. It also shows that the domain of [latex]f\circ g[/latex] can contain values that are not in the domain of [latex]f[/latex], though they must be in the domain of [latex]g[/latex].

Try It 6

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]
Solution

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