So, what is a 'relation'?
In math, a relation is just a set of ordered pairs.
Note:{ } are the symbol for "set"
Some Examples of Relations include { (0,1) , (55,22), (3,50) }
 { (0, 1) , (5, 2), (3, 9) }
 { (1,7) , (1, 7), (33, 7), (32, 7) }
One more time: A relation is just a set of ordered pairs. There is absolutely nothing special at all about the numbers that are in a relation. In other words, any bunch of numbers is a relation so long as these numbers come in pairs.
Video Lesson
What is the domain and range of a 'relation'?
Is the set of all the first numbers of the ordered pairs.
In other words, the domain is all of the xvalues.
Is the set of the second numbers in each pair, or the yvalues
Examples
of the Domain and Range
Example 2
Arrow Chart
Relations are often represented using arrow charts connecting the domain and range elements.
Example 3
Domain and range of a relation
In the relation above, the domain is { 2, 4, 11, 21}
the range is is { 5, 31, 11, 3}
I. Practice Identifying Domain and Range
Domain: 1, 2, 1, 8, 9
Range: 2, 51, 3, 22, 51
Domain: 5, 21, 11, 81, 19
Range: 6, 51, 93, 202, 51
Interactive Relation
See this demonstration by itself
What makes a relation a function?
Functions are a special kind of relation
At first glance, a function looks just like a relation.
 It's a set of ordered pairs such as { (0,1) , (5, 22), (11,9) }
 Like a relation, a function has a domain and range made up of the x and y values of ordered pairs.
In mathematics, what distinguishes a function from a relation is that each x value in a function has one and only ONE yvalue.
Since relation #1 has ONLY ONE y value for each x value, this relation is a function
On the other hand, relation #2 has TWO distinct y values 'a' and 'c' for the same x value of '5' . Therefore, relation #2 does not satisfy the definition of a mathematical function
Teachers has multiple students
If we put teachers into the domain and students into the range, we do not have a function because the same teacher, like Mr. Gino below, has more than 1 student in a classroom.
Mothers and Daughters Analogy
A way to try to understand this concept is to think of how mothers and their daughters could be represented as a function.
Each element in the domain, each daughter , can only have 1 mother (element in the range).
Some people find it helpful to think of the domain and range as people in romantic relationships. If each number in the domain is a person and each number in the range is a different person, then a function is when all of the people in the domain have 1 and only 1 boyfriend/girlfriend in the range.
Compare the two relations on the below. They differ by just one number, but only one is a function.
What's an easy way to do this?
II. Practice Identifying Functions
 Relation #1 { (1,2), (4,51), (1,2), (8,51) }
 Relation #2 { (13,14), (13,5), (16,7), (18,13) }
 Relation #3 { (3,90), (4,54), (6,71), (8,90) }
Relation #1 and Relation #3 are both functions.
 Relation #1{ (3,4), (4,5), (6,7), (8,9) }
 Relation #2 { (3,4), (4,5), (6,7), (3,9) }
 Relation #3{ (3,4), (4,5), (0,0), (8,9) }
 Relation #4 { (8, 11), (34,5), (6,17), (8,19) }
Relation #1 and Relation #3 are functions because each x value, each element in the domain, has one and only only one y value, or one and only number in the range.
 { (8 , 11), (34,5), (6,17), (X ,22) }
X cannot be 8, 34, or 6.
If x were 8 for instance, the relation would be:
{ (8, 11), (34,5), (6,17), (8 ,22) }
In this relation, the xvalue of 8 has two distinct y values.
Therefore this relation would NOT be a function since each element in the domain must have 1 and only value in the range.
 { ( 12, 13), (11, 22), (33, 101), (X ,22) }
X cannot be 12 or 33
If x were 12 for instance, the relation would be:
{ (12 , 13 ), (11 , 22), ( 33, 101), (12 ,22 }
In this problem, x could be 11. Since (11, 22) is already a pair in our relation, 11 can again go with a range element of 22 without creating a problem (We would just have two copies of 1 ordered pair).
If x were 11 , the relation would still be a function :
{ (12, 13 ), (11 , 22) , ( 33, 101), (11 ,22) }
The all important rule for a function in math  that each value in the domain has only 1 value in the range would still be true if we had a second copy of 1 ordered pair.
 { (12,14), (13,5) , (2,7), (X,13) }
X cannot be 12, 13, or 2.

Further Reading
 Functions Home
 Vertical Line Test
 One to one function
 Inverse of a Function
 Images