Math Functions, Relations, Domain & Range

From Vertex To Standard Form

Worksheet on Functions in Math
(relations, domain and range). This worksheet goes in tandem with this web page.

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

Examples

of the Domain and Range

Example 1
Picture of Domain and Range

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.

Arrow Chart
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

Problem 1

What is the domain and range of the following relation?

{ (-1,2), (2, 51), (1, 3), (8, 22), (9, 51) }

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

Range: 2, 51, 3, 22, 51

Problem 2

What is the domain and range of the following relation?

{ (-5,6), (21, -51), (11, 93), (81, 202), (19, 51) }

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

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

Problem 3

What is the domain and range of the following relation?

Domain and Range

Interactive Relation

See this demonstration by itself

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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.

Difference between relation and function

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.

teachers student arrow chart

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).

mother daughter function

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! repeat elements in the domain

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.

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.

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.

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