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)

**Solution**

The easiest way to figure out if a relation is a function is to look at all the

a) You can see that in this relation there are two different **not** a function.

b) Each value of

c) In this relation there are two different **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

### 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)
x−4−3−2−10y 169 410 - (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)
x0 1101001000y2−2 2−22 Age2025253035Number of jobs by that age3 4 7 4 2

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.15.