Relations And Functions Examples

In these lessons, will help Algebra 1 students learn how to distinguish between relations and functions and how to to solve real life problems that deal with relations.

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Relations And Functions
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Algebra Lessons

What Is A Relation?

A relation is any set of ordered pairs. An ordered-pair is a pair of values that go together. For example, (4, 7) is an ordered-pair number; the order is designated by the first element 4 and the second element 7.

What Is A Function?

  1. A function is a relation in which each x-element has only one y-element associated with it. Given a set of ordered pairs, a relation is a function if there are no repeated x-value.
  2. A relation is a function if there are no vertical lines that intersect its graph at more than one point. This is called the vertical line test.

Table of Values - One way to represent the relationship between the input and output variables in a relation or function is by means of a table of values.

Ordered Pairs - Relations and functions can also be represented as a set of points or ordered pairs.

Which of the following sets of ordered pairs represent functions?
A = {(0,-2), (1,4), (-3,3), (5,0)}
B = {(-4,0), (2,-3), (2,-5)}
C = {(-5,1), (2,1), (-3,1), (0,1)}
D = {(3,-4),(3,-2),(0,1),(2,-1)}
E = {(1,3)}

The Vertical Line Test

  • If all vertical lines intersect the graph of a relation in at most one point, the relation is also a function. One and only one output exists for each input.
  • If any vertical line intersects the graph of a relation at more than one point, the relation fails the test and is not a function. More than one value exists for some (or all) input value(s).

In general, we say that the output depends on the input.
Output variable = Dependent Variable
Input Variable = Independent Variable

If the relation is a function, then we say that the output is a function of the input.

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}

Relations And Functions

This video looks at relations and functions. It includes six examples of determining whether a relation is a function, using the vertical line test and by looking for repeated x values.


Learn about relations.


  1. Express the relation {(2,3),(4,7),(6,8)} as a table, as graph, and as a mapping diagram.
  2. Give the domain and range of the relation.

Relations - Problem Solving Applications

Learn to solve real life problems that deal with relations.

An electrician charges a base fee of $70 plus $50 for each hour of work. Create a table that shows the amount the electrician charges for 1,2,3, and 4 hours of work. Let x represent the number of hours and y represent the amount charged for x hours. Is this relation a function?


Learn about functions.

Give the domain and range of the relation. Tell whether the relation is a function.

  1. {(3,-2),(5,-1),(4,0),(3,1)}

Write Functions

Learn how do we write functions as rule.

Identify the independent and dependent variables. Write a rule in function notation for the situation.
a. A math tutor charges $35 per hour.
b. A fitness center charges a $100 initiation fee plus $40 per month.
c. Stephen buys lettuce that costs $1.69/lb.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
Mathway Calculator Widget

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