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학습 가이드 > MATH 1314: College Algebra

Find the domain of a composite function

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

A General Note: Domain of a Composite Function

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

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

Example 8: Finding the Domain of a Composite Function

Find the domain of

(fg)(x) wheref(x)=5x1 and g(x)=43x2\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(x)g\left(x\right)\\ consists of all real numbers except x=23x=\frac{2}{3}\\, since that input value would cause us to divide by 0. Likewise, the domain of ff consists of all real numbers except 1. So we need to exclude from the domain of g(x)g\left(x\right)\\ that value of xx for which g(x)=1g\left(x\right)=1\\.

{43x2=14=3x26=3xx=2\begin{cases}\frac{4}{3x - 2}=1\hfill \\ 4=3x - 2\hfill \\ 6=3x\hfill \\ x=2\hfill \end{cases}\\

So the domain of fgf\circ g is the set of all real numbers except 23\frac{2}{3}\\ and 22. This means that

x23orx2x\ne \frac{2}{3}\text{or}x\ne 2\\

We can write this in interval notation as

(,23)(23,2)(2,)\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

(fg)(x) wheref(x)=x+2 and g(x)=3x\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 gg is (,3]\left(-\infty ,3\right]\\. Now we check the domain of the composite function

(fg)(x)=3x+2 or(fg)(x)=5x\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 (,5]\left(-\infty ,5\right]\\. To find the domain of fgf\circ g\\, we ask ourselves if there are any further restrictions offered by the domain of the composite function. The answer is no, since (,3]\left(-\infty ,3\right]\\ is a proper subset of the domain of fgf\circ g\\. This means the domain of fgf\circ g\\ is the same as the domain of gg, namely, (,3]\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 fgf\circ g can contain values that are not in the domain of ff, though they must be in the domain of gg.

Try It 6

Find the domain of

(fg)(x) wheref(x)=1x2 and g(x)=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

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