﻿ Math Functions and Relations, what makes them different and how to Find the Domain and Range.

# 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) }

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'?

Step 1)
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.

Step 2)
The range:

Is the set of the second numbers in each pair, or the y-values

### Examplesof the Domain and Range

##### Example 1

In the relation above the domain is { 0, 3, 90 }
And the range is { 1, 22, 34 }

##### 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

#### What makes a relation a function?

Functions are a special kind of relation

At first glance, a function looks just like a relation.

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.

#### 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

• 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 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.

• { ( 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.

• { (12,14), (13,5) , (-2,7), (X,13) }

X cannot be 12, 13, or -2.

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