Relations and Functions

1
1-6
Relations and Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
2
Warm Up Use the graph for Problems 12.
1. List the x-coordinates of the points.
2. List the y-coordinates of
the points.
2, 0, 3, 5
3, 4, 1, 0
3
Objectives
Identify the domain and range of relations and
functions. Determine whether a relation is a
function.
4
Vocabulary
relation domain range function
5

• 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.
6
Mapping Diagram
Domain
Range
A
2
B
C
Set of Ordered Pairs
(2, A), (2, B), (2, C)
(x, y) (input, output) (domain,
range)
7
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.
8
Check It Out! Example 1
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.
9

• 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.

10
Number
Number, Letter
The numbers 6, 2, 8, and 4 each appear as the
first coordinate of three different
ordered pairs.
(6, M), (6, N), (6, O)
(2, A), (2, B), (2, C)
(8, T), (8, U), (8, V)
(4, G), (4, H), (4, I)
11

• 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.
12

• 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.

13
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.
14
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.
15
Check It Out! Example 2
Determine whether each relation is a function.
A.
There is only one price for each shoe size. The
relation from shoe sizes to price makes is a
function.
B. from the number of items in a grocery cart to
the total cost of the items in the cart
The number items in a grocery cart would be
associated with many different total costs of the
items in the cart. The relation of the number of
items in a grocery cart to the total cost of the
items is not a function.
16

• 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.

17
(No Transcript)
18
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.
19
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 not a function. A vertical line at x 1
would pass through (1, 1) and (1, 2).
20
Check It Out! Example 3a
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.
21
Check It Out! Example 3a
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).
22
Lesson Quiz Part I
1. Give the domain and range for this relation
(10, 5), (20, 5), (30, 5), (60, 100), (90,
100). Determine whether each relation is a
function. 2. from each person in class to the
number of pets he or she has 3. from city to
zip code
D 10, 20, 30, 60, 90) R 5, 100
function
not a function
23
Lesson Quiz Part II
Use the vertical-line test to determine whether
the relation is a function. If not, identify two
points a vertical line would pass through. 4.
not a function possible answer (3, 2) and (3,
2)