Math Functions, Relations, Domain & Range

Answer

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 .

The domain:	is the set of all the first numbers of the ordered pairs . In other words, the domain is all of the x-values.
The range:	is the set of the second numbers in each pair, or the y-values

Example 2 Arrow Chart

Relations are often represented using arrow charts connecting the domain and range elements.

Example 3 of 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}

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. 

Answer In mathematics, what distinguishes a function from a relation is that each x value in a function has one and only ONE y-value .

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.

Look for repeated elements in the domain

As soon as an element in the domain repeats, watch out!

Relation #1 and Relation #3 are both functions.

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.

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 x-value 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.

X cannot be 12 or 33 . If x were 12 for instance, the relation would be: { (12 , 13 ), (-11 , 22), ( 33, 101), (12 ,22 } Did we trick you? 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.

X cannot be 12, 13, or -2.