### Learning Outcomes

- Find the domain of a composite function.
- Decompose a composite function.

As we discussed previously, 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].

### A General Note: 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].

### Example: Finding the Domain of a Composite Function

Find the domain of

[latex]\left(f\circ g\right)\left(x\right)\text{ where}f\left(x\right)=\dfrac{5}{x - 1}\text{ and }g\left(x\right)=\dfrac{4}{3x - 2}[/latex]

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

Find the domain of

[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]

### Try It

Find the domain of

[latex]\left(f\circ g\right)\left(x\right)\text{ where}f\left(x\right)=\dfrac{1}{x - 2}\text{ and }g\left(x\right)=\sqrt{x+4}[/latex]

### Try It

We can use graphs to visualize the domain that results from a composition of two functions.

Graph the two functions below with an online graphing calculator.

- [latex]f(x)=\sqrt{3-x}[/latex]
- [latex]g(t) = \sqrt{x+4}[/latex]

Next, create a new function, [latex]h(x) = g(f(x))[/latex]. Based on the graph, what is the domain of this function? Explain why [latex]g(f(x))[/latex] and [latex]f(x)[/latex] have the same domain.

Now define another composition, [latex]p(x) = f(g(x)[/latex]. What is the domain of this function? Explain why you can evaluate [latex]g(10)[/latex], but not [latex]p(10)[/latex].

## Decompose a Composite Function

In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There is almost always more than one way to **decompose a composite function**, so we may choose the decomposition that appears to be most obvious.

### Example: Decomposing a Function

Write [latex]f\left(x\right)=\sqrt{5-{x}^{2}}[/latex] as the composition of two functions.

### Try It

Write [latex]f\left(x\right)=\dfrac{4}{3-\sqrt{4+{x}^{2}}}[/latex] as the composition of two functions.