Finding the Domain of a Composite Function

In this section, you will learn how to define the domain of a composite function.

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
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