1. Relations and Functions
June 21, 2012

2. What is a Relation? A relation is a set of ordered pairs.
When you group two or more points in a set, it is
referred to as a relation. When you want to show that a
set of points is a relation you list the points in braces.
For example, if I want to show that the points (-3,1) ;
(0, 2) ; (3, 3) ; & (6, 4) are a relation, it would be written
like this:
{(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)}

3. Domain and Range Each ordered pair has two parts, an x-value
and a y-value.
The x-values of a given relation are called the
Domain.
The y-values of the relation are called the
Range.
When you list the domain and range of
relation, you place each the domain and the
range in a separate set of braces.

4. For Example, {(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)}
1. List the domain and the range of the relation
{(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)}
Domain: { -3, 0, 3, 6} Range: {1, 2, 3, 4}
2. List the domain and the range of the relation
{(-3,3) ; (0, 2) ; (3, 3) ; (6, 4) ; ( 7, 7)}
Domain: {-3, 0, 3, 6, 7} Range: {3, 2, 4, 7}
Notice! Even though the number 3 is listed twice in the
relation, you only note the number once when you list the
domain or range!

5. A function is a relation that assigns each y-value only one x-value.
What is a Function?
A function is a relation that assigns each y-value only one x-value.
What does that mean? It means, in order for the
relation to be considered a function, there cannot be
any repeated values in the domain.
There are two ways to see if a relation is a function:
Vertical Line Test
Mappings

6. Using the Vertical Line Test
Use the vertical line test to check
if the relation is a function only if
the relation is already graphed.
Hold a straightedge (pen, ruler,
etc) vertical to your graph.
Drag the straightedge from left
to right on the graph.
3. If the straightedge intersects
the graph once in each spot ,
then it is a function.
If the straightedge intersects the
graph more than once in any
spot, it is not a function.
A function!

7. Examples of the Vertical Line Test
function
Not a function
Not a function
function
……….

8. It’s easier than it sounds 
Mappings
If the relation is not graphed, it is easier to use what is called a mapping.
When you are creating a mapping of a relation, you
draw two ovals.
In one oval, list all the domain values.
In the other oval, list all the range values.
Draw a line connecting the pairs of domain and range
values.
If any domain value ‘maps’ to two different range
values, the relation is not a function.
It’s easier than it sounds 

9. Example of a Mapping
Create a mapping of the following relation and state whether or not it is a function.
{(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)}
Steps
Draw ovals
List domain
List range
Draw lines to connect -3 3 6 1 2 3 4
This relation is a function because each x-value maps to only one y-value.

10. It is still a function if two x-values go to the same y-value.
Another Mapping
Create a mapping of the following relation and state whether or not it is a function.
{(-1,2) ; (1, 2) ; (5, 3) ; (6, 8)} -1 1 5 6
Notice that even
though there are
two 2’s in the
range, you only
list the 2 once. 2 3 8
This relation is a function because each x-value maps to only one y-value.
It is still a function if two x-values go to the same y-value.

11. It is NOT a function if one x-value go to two different y-values.
Last Mapping
Create a mapping of the following relation and state whether or not it is a function.
{(-4,-1) ; (-4, 0) ; (5, 1) ; (3, 9)}
Make sure to list
the (-4) only once! -1 1 9 -4 5 3
This relation is NOT a function because the (-4) maps to the (-1) & the (0).
It is NOT a function if one x-value go to two different y-values.
……….

12. Vocabulary Review Relation: a set of ordered pairs.
Domain: the x-values in the relation.
Range: the y-values in the relation.
Function: a relation where each x-value is assigned (maps to) on one y-value.
Vertical Line Test: using a vertical straightedge to see if the relation is a function.
Mapping: a diagram used to see if the relation is a function.
……….

13. Complete the following questions:
1. Identify the domain and range tell whether if the relation is a function or not:
a. {(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)}
b. {(0,-6) ; (1, 2) ; (7, -4) ; (1, 4)}
2. Use Vertical line test to tell whether if the relation is a function or not:
3. Use a mapping to see if the following relations are functions:
a. {(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)}

14. Answers (you will need to hit the spacebar to pull up the next slide)
1a. Domain: {-4, -2, 3, 4} Range: {-1, 2, 1}
1b. Domain: {0, 1, 7} Range: {-6, 2, -4, 4}
2a b.
3a b.
Function
Not a Function 1 7 -6 2 -4 4 -4 -2 3 4 -1 2 1
Function
Not a Function

15. Functions or Relations?
B={(3, -1), (3, -2), (1, -3), (0, -4)}
C={(-1, 2), (-2, 1), (-3, 0), (-4, -1)} 4. 5. 6. x -3 -1 1 2 5 6 y 7 9 10 11 12 17 16 x 4 16 36 49
100 y
± 2
± 4
± 6
± 7
± 10 x
± 3
± 4
± 5
± 6
± 8 y 10 17 16 37 65