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.