You just got a new part-time job at the mall that pays a base rate of $150/week plus $5/sale. Your boss encourages you to make as many sales as possible but she will cap your weekly earnings at $250. What are the domain and range of the function represented by this situation?

### Domain and Range of a Function

The input and output of a function is also called the domain and range. The **domain** of a function is the set of all input values. The **range** of a function is the set of all output values. Sometimes, a function is a set of points. In this case, the domain is all the

Let's determine if {(9, 2), (7, -3), (4, -6), (-10, 4), (-2, -7)} is a function. If so, we will find the domain and range.

First, this is a function because the

The

Let's find the domain and range for the following problems.

y=x−3

Because this is a linear equation we also know that it is a linear function. All lines continue forever in both directions, as indicated by the arrows.

Notice the line is solid, there are no dashes or breaks. This means that it is continuous. A **continuous** function has a value for every

Domain:

In words,

The second option, **interval notation.**

The range of this function is also continuous. Therefore, the range is also the set of all real numbers. We can write the range in the same ways we wrote the domain, but with

Range:

This is a function, even though it might not look like it. This type of function is called a **piecewise function** because it pieces together two or more parts of other functions.

To find the domain, look at the possible

Mathematically, this would be written:

However, upon further investigation, the branch on the left does pass through the yellow region, where we though the function was not defined. This means that the function is defined between 1 and -3 and thus for all real numbers. However, below -3, there are no

### Examples

#### Example 1

Earlier, you were asked to find the domain and range of the function of your sales, where you make a base rate of $150/week plus $5/sale (your weekly earnings are capped at $250/week).

The function represented by this situation can be written as *x* is the number of sales you make. You can't make a negative number of sales, so the least amount of sales you can make is zero. To find the maximum number of sales before you reach the cap, we must plug in $250 for *y*.

Therefore, the domain of the function is

To find the range, plug the two extremes of the domain into the equation. When *x* equals 0, *y* equals 150, and when *x* equals 20, *y* equals 250.

Therefore the range of the function is

#### Example 2

Find the domain and range of the following function: {(8, 3), (-4, 2), (-6, 1), (5, 7)}.

Domain:

#### Example 3

Find the domain and range of the following function:

Domain:

#### Example 4

Find the domain and range of the following function.

This is a piecewise function. The

Domain:

#### Example 5

Find the domain and range of the following function.

This is a parabola, the graph of a quadratic equation. Even though it might not look like it, the ends of the graph continue up, infinitely, and

Domain: \begin{align*}x \in \mathbb{R}\end{align*} Range: \begin{align*}y \in [-6,\infty)\end{align*}

### Review

Determine if the following sets of points are functions. If so, state the domain and range.

- {(5, 6), (-1, 5), (7, -3), (0, 9)}
- {(9, 8), (-7, 8), (-7, 9), (8, 8)}
- {(6, 2), (-5, 6), (-5, 2)}
- {(-1, 2), (-6, 3), (10, 7), (8, 11)}
- {(5, 7), (3, 7), (5, 8), (8, 1)}
- {(-3, -4), (-5, -6), (1, 2), (2, -6)}

Find the domain and range of the following functions.

- \begin{align*}y = 3x - 7\end{align*}
- \begin{align*}6x -2y = 10\end{align*}
**Challenge****Writing**Make a general statement about the domain and range of all linear functions. Use the proper notation.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.9.