# Relations and Functions Examples

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Videos, worksheets, solutions, and activities to help Algebra 1 students learn how to distinguish between relations and functions and how to to solve real life problems that deal with relations.

What is a relation?
A relation is any set of ordered pairs.

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.
Example:
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. 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.

Relations
Examples:
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
Examples:
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? Functions
Examples:
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.

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