### What Is a Composition of Functions?

A **function** is a relation (set of ordered pairs) in which each element of the domain corresponds to exactly one element of the range. The **domain of a function** is the set of first coordinates of the ordered pairs in the function. A value in the domain of a function is called the **input**. The **range of a function** is the set of second coordinates of the ordered pairs in the function. A value in the range of a function is called the **output**.

**composition of functions**. The composition of functions $f$ and $g$ is shown with symbols as:

The composition of two functions can be used to solve different types of problems, including conversion problems. If one function converts kilometers to meters and another converts meters to centimeters, then the composition converts kilometers to centimeters.

### Evaluating Composite Functions

The composition of two functions $f(g(x))$ is evaluated by first finding the value of $g(x)$ and then substituting that value into $f(x)$ and evaluating. This can be thought of as placing an input into a function to arrive at an output and then using that value as an input of a second function to get a final output.

For example:Use the graph to evaluate the inside function, $g(x)$, at $x=-1$ to find $g(-1)$.

Find the point on the graph of $g(x)$ where $x=-1$. Then determine the value of $g(x)$ at that point. The point is $(-1,2)$, so $g(-1)=2$.### Determining Composite Functions and Their Domains

The domain of $f(g(x))$ is all the values of $x$ that meet the conditions that:

- $x$ is in the domain of $g$.
- $g(x)$ is in the domain of $f$.

The domain of $f(g(x))$ may be all or only part of the domain of $g$. When a value in the domain of $g$ produces an output that is not in the domain of $f$, the value is not in the domain of $f(g(x))$.

To determine the domain of the composite function $f(g(x))$, write the composite function rule and find its domain. Then consider any values of $x$ that are not in the domain of $g$. A value of $x$ that makes $g(x)$ undefined must be excluded from the domain of $f(g(x))$.

For example, if $f(x)=\sqrt{x-8}$ and $g(x)=x-3$, first find the composite function $f(g(x))$. Then find its domain.There are often values that must be excluded from the domain of the composite function when the separate functions involve rational expressions with the variable in the denominator.

So the domain of $f(g(x))$ can be written as $x\neq0$.