Any connection between sets of values is a relation. Some relations are also functions. A function is a relation where each value in the first set is related to one, and only one, value in the second set.
Consider the relationship between each state in the United States and its capital city. Some states and capitals are listed in the table.
| State | Capital City |
|---|---|
| Maryland | Annapolis |
| California | Sacramento |
| Florida | Tallahassee |
| Texas | Austin |
| Iowa | Des Moines |
If someone were to name any of the 50 states, you could reply with the capital city because each state has one, and only one, capital city. The relationship between a state and its capital city is a function.
Is the relationship between the amount of time Ms. Kendall’s students studied and the quiz score they earned a function? Explain.
The relation, as ordered pairs, is (30, 84); (10, 52); (15, 66); (20, 84); (20, 61); (60, 100); (25, 97); (15, 80); (0, 72); (35, 90).
| Look at the ordered pairs. |
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A function relates each value in the first set to one, and only one, value in the second set. (30, 84); (10, 52); (15, 66); (20, 84); (20, 61); (60, 100); (25, 97); (15, 80); (0, 72); (35, 90). The values 15 and 20 are each related to two values in the second set. This relation is not a function. |
| You can also use the graph or the mapping diagram to show that this relation is not a function. |
Notice in the graph that the points (15, 66) and (15, 80) are vertically aligned. If you drew a vertical line through (15, 66), it would also pass through (15, 80). The same is true for the points (20, 84) and (20, 61). This indicates that the relation is not a function. 
In the mapping, you can clearly see that 15 and 20 are mapped to more than one number in the second set. 
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Only some relations are functions. Functions have special properties and are important in mathematics, so it is vital to recognize when a relation is a function.
Think you got it?
How well can you distinguish between relations and functions? Use the activity below to find out. Determine whether each relation is also a function, then click it to check your answer.
