## Domain of a Composition

### 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 $f\circ g$ 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.

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: Finding the Domain of a Composite Function

Find the domain of

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

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

Find the domain of

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

### Try It

Find the domain of

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

### 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 tool.

1. $f(x)=\sqrt{3-x}$
2. $g(x) = \sqrt{x+4}$

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

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

## 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 $f\left(x\right)=\sqrt{5-{x}^{2}}$ as the composition of two functions.

### Try It

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

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