1. Objectives Identify the domain and range of relations and functions.
Determine whether a relation is a function.
Vocabulary Words
Relation and Function
Vertical Line Test
Domain and Range

2. A relation is a pairing of input values with output values
A relation is a pairing of input values with output values. It can be shown as a set of ordered pairs (x,y), where x is an input and y is an output.
The set of input (x) values for a relation is called the domain, and the set of output (y) values is called the range.

3. Mapping Diagram
Domain
Range A 2 B C
Set of Ordered Pairs
{(2, A), (2, B), (2, C)}
(x, y) (domain, range)

4. Example 1: Identifying Domain and Range
Give the domain and range for this relation:
{(100,5), (120,5), (140,6), (160,6), (180,12)}.
List the set of ordered pairs:
{(100, 5), (120, 5), (140, 6), (160, 6), (180, 12)}
Domain: {100, 120, 140, 160, 180}
The set of x-coordinates.
Range: {5, 6, 12}
The set of y-coordinates.

5. Give the domain and range for the relation shown in the graph.
Example 2
Give the domain and range for the relation shown in the graph.
List the set of ordered pairs:
{(–2, 2), (–1, 1), (0, 0),
(1, –1), (2, –2), (3, –3)}
Domain: {–2, –1, 0, 1, 2, 3}
The set of x-coordinates.
Range: {–3, –2, –1, 0, 1, 2}
The set of y-coordinates.

6. Not a function: The relationship from number to letter is not a function because the domain value 2 is mapped to the range values A, B, and C.
Function: The relationship from letter to number is a function because each letter in the domain is mapped to only one number in the range.
Both functions and non-functions are relations!

8. Example 3A: Using the Vertical-Line Test
Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.
This is a function. Any vertical line would pass through only one point on the graph.

9. Example 3B Using the Vertical-Line Test
Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.
This is a function. Any vertical line would pass through only one point on the graph.

10. Example 3C: Using the Vertical-Line Test
Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.
This is not a function. A vertical line at x = 1 would pass through (1, 1) and (1, –2).

11. Example 4A: Is this relation a function?
Function: The relationship is a function if each member of the domain is mapped to only one number in the range.
Example 4A: Is this relation a function?
From graph or ordered pairs:
{(–2, 2), (–1, 1), (0, 0),
(1, –1), (2, –2), (3, –3)}

12. Example 4B: Is this relation a function?
Function: The relationship is a function if each member of the domain is mapped to only one number in the range.
Example 4B: Is this relation a function?
{(100,5), (120,5), (140,6), (160,6), (180,12)}.
Identify the domain and range:
Domain: {100, 120, 140, 160, 180}
Range: {5, 6, 12}

13. Example 4C: Is this relation a function?
There is only one price for each shoe size. The relation from shoe sizes to price makes is a function.
B. Names in a classroom and grades
C. (Reverse B…) from class grades, find student name.

14. Example 5A: Is this relation a function?
Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.
This is not a function. A vertical line at x = 1 would pass through (1, 2) and (1, –2).

15. Example 5B: Is this relation a function?
Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.
not a function