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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, and determining 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?
A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.
The graph of a function f is a drawing hat represents all the input-output pairs, (x, f(x)). In cases where the function is given by an equation, the graph of a function is the graph of the equation y = f(x).
The vertical line test - a graph represents a function if it is impossible to draw a vertical line that intersects the graph more than once.
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.

1. Using a mapping diagram, determine whether each relation is a function.
2. Using a vertical line test, determine whether the relation is a function.
3. Make a table for f(t) = 0.5x + 1. Use 1, 2, 3, and 4 as domain values.
4. Evaluate the function rule f(g) = -2g + 4 to find the range for the domain (-1, 3, 5). Determine if a Relation is a Function
This video explains the concepts behind mapping a relation and the vertical line test. Relations and Functions

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