Finding the Domain of a Composite Function

As we discussed previously, the domain of a composite function such as \(f \circ g\) is dependent on the domain of \(g\) and the domain of \(f\). 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 \(f \circ g\). Let us assume we know the domains of the functions \(f\) and \(g\) separately. If we write the composite function for an input \(x\) as \(f(g(x))\), we can see right away that \(x\) must be a member of the domain of \(g\) in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that \(g(x)\) must be a member of the domain of \(f\), otherwise the second function evaluation in \(f(g(x))\) cannot be completed, and the expression is still undefined. Thus the domain of \(f \circ g\) consists of only those inputs in the domain of \(g\) that produce outputs from \(g\) belonging to the domain of \(f\). Note that the domain of \(f\) composed with \(g\) is the set of all \(x\) such that \(x\) is in the domain of \(g\) and \(g(x)\) is in the domain of \(f .\)

DOMAIN OF A COMPOSITE FUNCTION

The domain of a composite function \(f(g(x))\) is the set of those inputs \(x\) in the domain of \(g\) for which \(g(x)\) is in the domain of \(f\).

HOW TO

Given a function composition \(f(g(x))\), 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 \(f \circ g\).

EXAMPLE 8

Finding the Domain of a Composite Function

Find the domain of

\((f \circ g)(x) \, \text{where} \qquad f(x)=\frac{5}{x-1} \qquad \text { and } \qquad g(x)=\frac{4}{3 x-2} \)

Solution

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

\(

\begin{aligned}

\frac{4}{3 x-2} &=1 \\

4 &=3 x-2 \\

6 &=3 x \\

x &=2

\end{aligned}

\)

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

\(x \neq \frac{2}{3} \quad \text { or } \quad x \neq 2\)

We can write this in interval notation as

\(\left(-\infty, \frac{2}{3}\right) \cup\left(\frac{2}{3}, 2\right) \cup(2, \infty)\)

EXAMPLE 9

Finding the Domain of a Composite Function Involving Radicals

Find the domain of

\((f \circ g)(x) \quad \text { where } \quad f(x)=\sqrt{x+2} \text { and } \quad g(x)=\sqrt{3-x}\)

Solution

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

\((f \circ g)(x)=\sqrt{\sqrt{3-x}+2}\)

For \((f \circ g)(x)=\sqrt{\sqrt{3-x}+2}, \sqrt{3-x}+2 \geq 0\), since the radicand of a square root must be positive. Since square roots are positive, \(\sqrt{3-x} \geq 0\), or, \(3-x \geq 0\), which gives a domain of \((-\infty, 3]\).

Analysis

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 \(f \circ g\) can contain values that are not in the domain of \(f\), though they must be in the domain of \(g\).

TRY IT #6

Find the domain of

\((f \circ g)(x) \quad \text{where} \qquad f(x)=\frac{1}{x-2} \qquad \text { and } \qquad g(x)=\sqrt{x+4}\)

Source: Rice University, https://openstax.org/books/college-algebra/pages/3-4-composition-of-functions
This work is licensed under a Creative Commons Attribution 4.0 License.

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 10

Decomposing a Function

Write \(f(x)=\sqrt{5-x^{2}}\) as the composition of two functions.

Solution

We are looking for two functions, \(g\) and \(h\), so \(f(x)=g(h(x))\). To do this, we look for a function inside a function in the formula for \(f(x)\). As one possibility, we might notice that the expression \(5-x^{2}\) is the inside of the square root. We could then decompose the function as

\(h(x)=5-x^{2} \quad \text { and } g(x)=\sqrt{x}\)

We can check our answer by recomposing the functions.

\(g(h(x))=g\left(5-x^{2}\right)=\sqrt{5-x^{2}}\)

TRY IT #7

Write \(f(x)=\dfrac{4}{3-\sqrt{4+x^{2}}}\) as the composition of two functions.