1. Algebra 2 1.6 Relations and Functions
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 values for a relation is called the domain, and the set of output values is called the range.
Mapping Diagram
Domain
Range A B C 2
Set of Ordered Pairs: {(2, A), (2, B), (2, C)}
(x, y) (input, output) (domain, range)

2. Algebra 2 1.6 Relations and Functions
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.

3. Algebra 2 1.6 Relations and Functions
Suppose you are told that a person entered a word into a text message using the numbers 6, 2, 8, and 4 on a cell phone. It would be difficult to determine the word without seeing it because each number can be used to enter three different letters.
However, if you are told to enter the word MATH into a text message,
you can easily determine that you use the numbers 6, 2, 8, and 4,
because each letter appears on only one numbered key.
The first coordinate is different in each ordered pair.
{(M, 6), (A, 2), (T, 8), (H,4)}
A relation in which the first coordinate is never repeated is called a function. In a function, there is only one output for each input, so each element of the domain is mapped to exactly one element in the range.
Although a single input in a function cannot be mapped to more than one output, two or more different inputs can be mapped to the same output.

4. Algebra 2 1.6 Relations and Functions
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.

5. Algebra 2 1.6 Relations and Functions
Example 2: Determining Whether a Relation is a Function
Determine whether each relation is a function.
A. from the items in a store to their prices on a certain date
There is only one price for each different item on a certain date. The relation from items to price makes it a function.
B. from types of fruits to their colors
A fruit, such as an apple, from the domain would be associated with more than one color, such as red and green. The relation from types of fruits to their colors is not a function.

6. Algebra 2 1.6 Relations and Functions
Every point on a vertical line has the same x-coordinate, so a vertical line cannot represent a function. If a vertical line passes through more than one point on the graph of a relation, the relation must have more than one point with the same x-coordinate. Therefore the relation is not a function.

7. Algebra 2 1.6 Relations and Functions
Ex 3:
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.
This is not a function. A vertical line at x = 1 would pass through (1, 1) and (1, –2).