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Relations and Functions

In these lessons, we will look at ordered-pair numbers, relations and an introduction to functions.

Related Topics: More Algebra Lessons

Ordered-Pair Numbers

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.











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.


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)}


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.

Determine if a Relation is a Function
A Function is a correspondence between a first set, called the domain, and a second set, called the range, such that eah member of the domain corresponds to exactly one menber of the range.

How to Determine if a Relation is a Function
This Algebra 1 level math video tutorial
- defines a relation as a set of ordered pairs and a function as a relation with one to one correspondence
- models how to determine if a relation is a function with two different methods
- shows how to use a mapping and the vertical line test.
- discusses how to work with function notation. It is defined as replacing y in an equation that is a function.
Relations and Functions.

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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