Find 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 $f\left(g\left(x\right)\right)$ , 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 $g\left(x\right)$ must be a member of the domain of $f$ , otherwise the second function evaluation in $f\left(g\left(x\right)\right)$ 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 $g\left(x\right)$ is in the domain of $f$ .

A General Note: Domain of a Composite Function

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

How To: Given a function composition $f\left(g\left(x\right)\right)$ , 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 $f\circ g$ .

Example 8: Finding the Domain of a Composite Function

Find the domain of

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

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

Solution

The domain of $g\left(x\right)$ 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 $g\left(x\right)$ that value of $x$ for which $g\left(x\right)=1$ .

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{4}{3 x - 2} = 1 4 = 3 x - 2 6 = 3 x x = 2 \end{matrix}

$\begin{cases}\frac{4}{3x - 2}=1\hfill \\ 4=3x - 2\hfill \\ 6=3x\hfill \\ x=2\hfill \end{cases}$

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} or x \neq 2

$x\ne \frac{2}{3}\text{or}x\ne 2$

We can write this in interval notation as

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

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

Example 9: Finding the Domain of a Composite Function Involving Radicals

Find the domain of

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

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

Solution

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

(f \circ g) (x) = \sqrt{3 - x + 2} or (f \circ g) (x) = \sqrt{5 - x}

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

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

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

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

Solution

Composition of Functions