What if you were given a set of *x* and *y* values? How could you determine whether the relation between those values represented a function? After completing this Concept you'll be able to analyze the domain and range of a relation to determine if it represents a function.

### Watch This

CK-12 Foundation: 0115S Relations and Functions

### Guidance

A function is a special kind of **relation**. In a function, for each input there is exactly one output; in a relation, there can be more than one output for a given input.

#### Example A

Consider the relation that shows the heights of all students in a class. The domain is the set of people in the class and the range is the set of heights. This relation is a function because each person has exactly one height. If any person had more than one height, the relation would not be a function.

Notice that even though the same person can’t have more than one height, it’s okay for more than one person to have the same height. In a function, more than one input can have the same output, as long as more than one output never comes from the same input.

#### Example B

*Determine if the relation is a function.*

a) (1, 3), (-1, -2), (3, 5), (2, 5), (3, 4)

b) (-3, 20), (-5, 25), (-1, 5), (7, 12), (9, 2)

c) \begin{align*}& x \quad 2 \quad \ \ 1 \quad \ 0 \quad 1 \quad 2\\ & y \quad 12 \quad 10 \quad 8 \quad 6 \quad 4\end{align*}

**Solution**

The easiest way to figure out if a relation is a function is to look at all the \begin{align*}x-\end{align*}values in the list or the table. If a value of \begin{align*}x\end{align*} appears more than once, and it’s paired up with different \begin{align*}y-\end{align*}values, then the relation is not a function.

a) You can see that in this relation there are two different \begin{align*}y-\end{align*}values paired with the \begin{align*}x-\end{align*}value of 3. This means that this relation is **not** a function.

b) Each value of \begin{align*}x\end{align*} has exactly one \begin{align*}y-\end{align*}value. The relation is a function.

c) In this relation there are two different \begin{align*}y-\end{align*}values paired with the \begin{align*}x-\end{align*}value of 2 and two \begin{align*}y-\end{align*}values paired with the \begin{align*}x-\end{align*}value of 1. The relation is **not** a function.

When a relation is represented graphically, we can determine if it is a function by using the **vertical line test**. If you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function.

#### Example C

*For the following graphs, determine whether they are functions.*

**Solution:**

1. Not a function. It fails the vertical line test.

2. A function. No vertical line will cross more than one point on the graph.

Watch this video for help with the Examples above.

CK-12 Foundation: Relations and Functions

### Vocabulary

- A
**function**is a special kind of**relation**. In a function, for each input there is exactly one output; in a relation, there can be more than one output for a given input.

### Guided Practice

*For the following graphs, determine whether they are functions.*

**Solution:**

1. A function. No vertical line will cross more than one point on the graph.

2. Not a function. It fails the vertical line test.

### Explore More

In 1-8, determine whether each relation is a function:

- (1, 7), (2, 7), (3, 8), (4, 8), (5, 9)
- (1, 1), (1, -1), (4, 2), (4, -2), (9, 3), (9, -3)
- \begin{align*}& x \quad -4 \quad -3 \quad -2 \quad -1 \quad 0\\ & y \quad \ \ 16 \qquad 9 \qquad \ 4 \qquad 1 \quad 0\end{align*}
- (2, -6), (1, -3), (0, 0), (1, 3), (2, 6)
- (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8)
- (-5, 10), (-1, 5), (0, 10), (1, 5), (5, 10)
- \begin{align*}& x \quad 0 \quad \ \quad 1 \quad 10 \quad 100 \quad 1000\\ & y \quad 2 \quad -2 \quad \ 2 \quad -2 \qquad 2\end{align*}
- \begin{align*}& \text{Age} \qquad \qquad \qquad \qquad \qquad \quad 20 \quad 25 \quad 25 \quad 30 \quad 35\\ & \text{Number of jobs by that age} \quad 3 \quad \ 4 \quad \ 7 \quad \ \ 4 \quad \ 2\end{align*}

In 9-10, use the vertical line test to determine whether each relation is a function.