The domain of a composite function such as [latex]f\circ g[/latex] is dependent on the domain of [latex]g[/latex] and the domain of [latex]f[/latex]. 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 [latex]f\circ g[/latex]. Let us assume we know the domains of the functions [latex]f[/latex] and [latex]g[/latex] separately. If we write the composite function for an input [latex]x[/latex] as [latex]f\left(g\left(x\right)\right)[/latex], we can see right away that [latex]x[/latex] must be a member of the domain of [latex]g[/latex] in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that [latex]g\left(x\right)[/latex] must be a member of the domain of [latex]f[/latex], otherwise the second function evaluation in [latex]f\left(g\left(x\right)\right)[/latex] cannot be completed, and the expression is still undefined. Thus the domain of [latex]f\circ g[/latex] consists of only those inputs in the domain of [latex]g[/latex] that produce outputs from [latex]g[/latex] belonging to the domain of [latex]f[/latex]. Note that the domain of [latex]f[/latex] composed with [latex]g[/latex] is the set of all [latex]x[/latex] such that [latex]x[/latex] is in the domain of [latex]g[/latex] and [latex]g\left(x\right)[/latex] is in the domain of [latex]f[/latex].
domain of a composite function
The domain of a composite function [latex]f\left(g\left(x\right)\right)[/latex] is the set of those inputs [latex]x[/latex] in the domain of [latex]g[/latex] for which [latex]g\left(x\right)[/latex] is in the domain of [latex]f[/latex].
How To: Given a function composition [latex]f\left(g\left(x\right)\right)[/latex], determine its domain.
Find the domain of [latex]g[/latex].
Find the domain of [latex]f[/latex].
Find those inputs, [latex]x[/latex], in the domain of [latex]g[/latex] for which [latex]g(x)[/latex] is in the domain of [latex]f[/latex]. That is, exclude those inputs, [latex]x[/latex], from the domain of [latex]g[/latex] for which [latex]g(x)[/latex] is not in the domain of [latex]f[/latex]. The resulting set is the domain of [latex]f\circ g[/latex].
Find the domain of
[latex]\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}[/latex]
The domain of [latex]g\left(x\right)[/latex] consists of all real numbers except [latex]x=\frac{2}{3}[/latex], since that input value would cause us to divide by 0. Likewise, the domain of [latex]f[/latex] consists of all real numbers except 1. So we need to exclude from the domain of [latex]g\left(x\right)[/latex] that value of [latex]x[/latex] for which [latex]g\left(x\right)=1[/latex].
[latex]\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}[/latex]
Because we cannot take the square root of a negative number, the domain of [latex]g[/latex] is [latex]\left(-\infty ,3\right][/latex]. Now we check the domain of the composite function
[latex]\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}[/latex]
The domain of this function is [latex]\left(-\infty ,5\right][/latex]. To find the domain of [latex]f\circ g[/latex], we ask ourselves if there are any further restrictions offered by the domain of the composite function. The answer is no, since [latex]\left(-\infty ,3\right][/latex] is a proper subset of the domain of [latex]f\circ g[/latex]. This means the domain of [latex]f\circ g[/latex] is the same as the domain of [latex]g[/latex], namely, [latex]\left(-\infty ,3\right][/latex].
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 [latex]f\circ g[/latex] can contain values that are not in the domain of [latex]f[/latex], though they must be in the domain of [latex]g[/latex].
Decomposing a Composite Function
Sometimes a function is easier to work with if we break it into simpler pieces. That means writing the function as a composition of two simpler functions. Because there can be more than one way to do this, we usually choose the decomposition that makes the problem easiest to handle.
Write [latex]f(x)=\sqrt{5-{x}^{2}}[/latex] as the composition of two functions. We are looking for two functions, [latex]g[/latex] and [latex]h[/latex], so [latex]f\left(x\right)=g\left(h\left(x\right)\right)[/latex]. To do this, we look for a function inside a function in the formula for [latex]f\left(x\right)[/latex].There are multiple ways to express [latex]f(x)=\sqrt{5-{x}^{2}}[/latex] as the composition of two functions.