An ordered-pair number is a pair of numbers that go together. The numbers are written within a set of parentheses and separated by a comma.
For example, (4, 7) is an ordered-pair number; the order is designated by the first element 4 and the second element 7. The pair (7, 4) is not the same as (4, 7) because of the different ordering. Sets of ordered-pair numbers can represent relations or functions.
A relation is any set of ordered-pair numbers.
Suppose the weights of four students are shown in the following table.
Student |
1 |
2 |
3 |
4 |
Weight |
120 |
100 |
150 |
130 |
The pairing of the student number and his corresponding weight is a relation and can be written as a set of ordered-pair numbers.
W = {(1, 120), (2, 100), (3, 150), (4, 130)}
The set of all first elements is called the domain of the relation.
The domain of W = {1, 2, 3, 4}
The set of second elements is called the range of the relation.
The range of W = {120, 100, 150, 130}
A function is a relation in which no two ordered pairs have the same first element.
A function associates each element in its domain with one and only one element in its range.
Example:
Determine whether the following are functions
a) A = {(1, 2), (2, 3), (3, 4), (4, 5)}
b) B = {(1, 3), (0, 3), (2, 1), (4, 2)}
c) C = {(1, 6), (2, 5), (1, 9), (4, 3)}
Solution:
a) A = {(1, 2), (2, 3), (3, 4), (4, 5)} is a function because all the first elements are different.
b) B = {(1, 3), (0, 3), (2, 1), (4, 2)} is a function because all the first elements are different. (The second element does not need to be unique)
c) C = {(1, 6), (2, 5), (1, 9), (4, 3)} is not a function because the first element, 1, is repeated.
A function can be identified from a graph. If any vertical line drawn through the graph cuts the graph at more than one point, then the relation is not a function. This is called the vertical line test.
Determining Whether a Relation is a Function
Understanding relations (defined as a set of inputs and corresponding outputs) is an important step to learning what makes a function. A function is a specific relation, anddetermining whether a relation is a function is a skill necessary for knowing what we can graph. Determining whether a relation is a function involves making sure that for every input there is only one output.
How to Determine if a Relation is a Function
This video explains the concepts behind mapping a relation and the vertical line test.