Math Functions, Relations, Domain & Range

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)}
• {(-1, 7)}
Non Examples of Relations i
• { 3, 1, 2 }
• {(0, 1, 2 ) , (3,4,5)} ( these numbers are grouped as 3's so not ordered and therefore not a relation )
• {-1, 7, 3,4,5,5}

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'?

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 1

In the relation above the domain is { 5, 1 , 3 } .( highlight )
And the range is {10, 20, 22} ( highlight ).

Example 2

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}.

Example 3

Arrow Chart

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

I. Practice Identifying Domain and Range

Problem 1

Domain: -1, 2, 1, 8, 9

Range: 2, 51, 3, 22, 51

Problem 2

Domain: -5, 21, 11, 81, 19

Range: 6, -51, 93, 202, 51

What makes a relation a function?

Functions are a special kind of relation.

At first glance, a function looks like a relation.

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.

What's an easy way to do this?

Look for repeated elements in the domain. As soon as an element in the domain repeats, watch out!

II. Practice Identifying Functions

Problem 1

Which relations below are 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. Remember if domain element repeats then it's not a function.

Problem 2

Which relations below are 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.

Remember if a domain element repeats then it's not a function.

Practice 3

For the following relation to be a function, X can not be what values?

• {(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 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.

Practice 4

For the relation below to be a function, X cannot be what values?

• {(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}

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.

Practice 5

For the relation below to be a function, X cannot be what values?

• {(12,14), (13,5) , (-2,7), (X,13)}

X cannot be 12, 13, or -2.