The Functions

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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) }

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 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.

Examples of the Domain and Range

 Examples of the domain and range of a relation.
 
In the relation above the domain is { 0, 3, 90 }

And the range is { 1, 22, 34 } 

Example two of the domain and range of a relation

In the relation above,

the domain is { 2, 4, 11, -21} the range is { -5, 31, -11, 3}

 Practice Identifying Domain and Range 

What is the domain and range of the following relation?
{ (-1,2), (2, 51), (1, 3), (8, 22), (9, 51) }

Answer

Domain: -1, 2, 1, 8, 9
Range: 2, 51, 3, 22, 51

What is the domain and range of the following relation
{ (-5,6), (21, -51), (11, 93), (81, 202), (19, 51) }

Answer

Domain: -5, 21, 11, 81, 19
Range: 6, -51, 93, 202, 51

What is the domain and range of the following relation?

Answer

  Interactive Relation

As the ball drops, you can see the relation of the ball's position and time.
The general formula to express this relation is

f(time)= position

Try changing the speed of ball's position. How does the speed effect the relation.
What makes a relation a function in Math?

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 y-value.

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.
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 '2'  and '4' for the same x value of '1'. Therefore, relation #2 does not satisfy the definition of a mathematical function.

Practice Identifying Functions

  Practice Problem one

Which relations below arefunctions?
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) }

    The Functions   
Relation #1 and Relation #3 are both functions.

Practice Problem Two

Which relations below arefunctions?

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) }
    The 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.

Practice Problem Three) For the following relation to be a function, X can not be what values?

{ (8, 11), (34,5), (6,17), (X ,22) }

    Answer    

X cannot be 8, 34, or 6.

If x was 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 and therefore this relation would NOT be a function since each element in the domain of a function in math must have 1 and only value in the range. 

 Practice Problem Four) For the relation below to be a function, X cannot be what values?

{ (12, 13), (-11, 22), (33, 101), (X ,22) }

    Answer    

X cannot be 12 or 33 .

If x was 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). The all important rule for a function in math -- that each value in the domain have only 1 value in the range-- would still be true if we had a second copy of 1 ordered pair.

 Practice Problem Five) For the relation below to be a function, X cannot be what values?

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

    Answer    

X cannot be 12, 13, or -2.

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