1-6
1-6 Relations
Relations and
and Functions
Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
1-6
Relations and Functions
Warm Up
Use the graph for Problems 1–2.
1. List the x-coordinates of the points.
–2, 0, 3, 5
2. List the y-coordinates of the points.
3, 4, 1, 0
Holt Algebra 2
1-6
Relations and Functions
Objectives
Identify the domain and range of relations
and functions.
Determine whether a relation is a function.
Holt Algebra 2
1-6
Relations and Functions
Vocabulary
relation
domain
range
function
Holt 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.
Holt Algebra 2
1-6
Relations and Functions
Mapping Diagram
Domain
Range
A
2
B
C
Set of Ordered Pairs
{(2, A), (2, B), (2, C)}
(x, y)
Holt Algebra 2
(input, output)
(domain, range)
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}
Holt Algebra 2
The set of y-coordinates.
1-6
Relations and Functions
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.
Holt 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.
Holt Algebra 2
1-6
Relations and Functions
Number
{Number, Letter}
{(6, M), (6, N), (6, O)}
{(2, A), (2, B), (2, C)}
{(8, T), (8, U), (8, V)}
{(4, G), (4, H), (4, I)}
Holt Algebra 2
The numbers 6, 2, 8,
and 4 each appear as
the first coordinate of
three different ordered
pairs.
1-6
Relations and Functions
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.
{(M, 6), (A, 2), (T, 8), (H,4)}
The first coordinate is different
in each ordered pair.
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.
Holt Algebra 2
1-6
Relations and Functions
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.
Holt 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.
Holt 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.
Holt Algebra 2
1-6
Relations and Functions
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.
Holt 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 xcoordinate. Therefore the relation is not a
function.
Holt Algebra 2
1-6
Relations and Functions
Holt Algebra 2
1-6
Relations and Functions
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.
Holt Algebra 2
1-6
Relations and Functions
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).
Holt Algebra 2
1-6
Relations and Functions
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.
Holt Algebra 2
1-6
Relations and Functions
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).
Holt Algebra 2
1-6
Relations and Functions
Lesson Quiz: Part I
1. Give the domain and range for this relation:
{(10, 5), (20, 5), (30, 5), (60, 100), (90, 100)}.
D: {10, 20, 30, 60, 90)}
R: {5, 100}
Determine whether each relation is a function.
2. from each person in class to the number of pets
he or she has function
3. from city to zip code not a function
Holt Algebra 2
1-6
Relations and Functions
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)
Holt Algebra 2