4-2 Relations and Functions
Objectives
Identify functions.
Find the domain and range of relations and
functions.
Holt Algebra 1
4-2 Relations and Functions
Relationships can be represented by a set of
ordered pairs called a relation.
In the scoring systems of some track meets, for
first place you get 5 points, for second place you
get 3 points, for third place you get 2 points, and
for fourth place you get 1 point. This scoring
system is a relation, so it can be shown by
ordered pairs. {(1, 5), (2, 3), (3, 2) (4, 1)}. You
can also show relations in other ways, such as
tables, graphs, or mapping diagrams.
Holt Algebra 1
4-2 Relations and Functions
Example 1: Showing Multiple Representations of
Relations
Express the relation {(2, 3), (4, 7), (6, 8)} as a
table, as a graph, and as a mapping diagram.
Table
x
y
2
3
4
7
6
8
Holt Algebra 1
Write all x-values under “x” and all
y-values under “y”.
4-2 Relations and Functions
Example 1 Continued
Express the relation {(2, 3), (4, 7), (6, 8)} as a
table, as a graph, and as a mapping diagram.
Graph
Use the x- and y-values to
plot the ordered pairs.
Holt Algebra 1
4-2 Relations and Functions
Example 1 Continued
Express the relation {(2, 3), (4, 7), (6, 8)} as a
table, as a graph, and as a mapping diagram.
Mapping Diagram
Holt Algebra 1
x
y
2
3
4
7
6
8
Write all x-values under “x” and
all y-values under “y”. Draw an
arrow from each x-value to its
corresponding y-value.
4-2 Relations and Functions
Check It Out! Example 1
Express the relation {(1, 3), (2, 4), (3, 5)} as
a table, as a graph, and as a mapping diagram.
Table
x
y
Write all x-values under “x” and all
y-values under “y”.
Holt Algebra 1
4-2 Relations and Functions
Check It Out! Example 1 Continued
Express the relation {(1, 3), (2, 4), (3, 5)} as
a table, as a graph, and as a mapping diagram.
Graph
Use the x- and y-values to
plot the ordered pairs.
Holt Algebra 1
4-2 Relations and Functions
Check It Out! Example 1 Continued
Express the relation {(1, 3), (2, 4), (3, 5)} as
a table, as a graph, and as a mapping diagram.
Mapping Diagram
y
x
Write all x-values under “x” and
all y-values under “y”. Draw an
arrow from each x-value to its
corresponding y-value.
Holt Algebra 1
4-2 Relations and Functions
The domain of a relation is the set of first
coordinates (or x-values) of the ordered
pairs. The range of a relation is the set of
second coordinates (or y-values) of the
ordered pairs. The domain of the track meet
scoring system is {1, 2, 3, 4}. The range is
{5, 3, 2, 1}.
The domain and range are in alphabetical
order. (X,Y) is the same as (D,R)
Holt Algebra 1
4-2 Relations and Functions
Example 2: Finding the Domain and Range of a
Relation
Give the domain and range of the relation.
The domain value is all x-values
from 1 through 5, inclusive.
The range value is all y-values
from 3 through 4, inclusive.
Domain: 1 ≤ x ≤ 5
Range: 3 ≤ y ≤ 4
Holt Algebra 1
4-2 Relations and Functions
Check It Out! Example 2a
Give the domain and range of the relation.
6
5
2
1
Domain:
Range:
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–4
–1
0
The domain values are all
x-values 1, 2, 5 and 6.
The range values are
y-values 0, –1 and –4.
4-2 Relations and Functions
Check It Out! Example 2b
Give the domain and range of the relation.
x
y
1
1
4
4
8
1
Domain:
Range:
Holt Algebra 1
The domain values are all
x-values 1, 4, and 8.
The range values are
y-values 1 and 4.
4-2 Relations and Functions
A function is a special type of relation
that pairs each domain value with exactly
one range value.
Which means…
Each X can be paired with
only one Y!!
Holt Algebra 1
4-2 Relations and Functions
Example 3A: Identifying Functions
Give the domain and range of the relation. Tell
whether the relation is a function. Explain.
{(3, –2), (5, –1), (4, 0), (3, 1)}
D:
R:
The relation is not a function. Each domain value
does not have exactly one range value. The domain
value 3 is paired with the range values –2 and 1.
Holt Algebra 1
4-2 Relations and Functions
Example 3B: Identifying Functions
Give the domain and range of the relation. Tell
whether the relation is a function. Explain.
–4
–8
4
5
D:
2
1
R:
This relation is a function. Each domain value is
paired with exactly one range value.
Holt Algebra 1
4-2 Relations and Functions
Example 3C: Identifying Functions
Give the domain and range of the relation. Tell
whether the relation is a function. Explain.
Draw in lines to
see the domain
and range
values
D:
R:
The relation is not a function. Nearly all domain
values have more than one range value.
Holt Algebra 1
4-2 Relations and Functions
Check It Out! Example 3
Give the domain and range of each relation. Tell
whether the relation is a function and explain.
a. {(8, 2), (–4, 1),
(–6, 2),(1, 9)}
b.
D:
R:
D:
R:
Holt Algebra 1
4-2 Relations and Functions
Graphs can be used to illustrate many different
situations. For example, trends shown on a
cardiograph can help a doctor see how a
patient’s heart is functioning.
To relate a graph to a given situation, use key
words in the description.
Holt Algebra 1
4-2 Relations and Functions
Example 1: Relating Graphs to Situations
Each day several leaves fall from a tree. One
day a gust of wind blows off many leaves.
Eventually, there are no more leaves on the
tree. Choose the graph that best represents
the situation.
Step 1 Read the graphs from left to right to show
time passing.
Holt Algebra 1
4-2 Relations and Functions
Check It Out! Example 1
The air temperature increased steadily for
several hours and then remained constant. At
the end of the day, the temperature increased
slightly before dropping sharply. Choose the
graph that best represents this situation.
Step 1 Read the graphs from left to right to show
time passing .
Holt Algebra 1
4-2 Relations and Functions
As seen in Example 1, some graphs are
connected lines or curves called continuous
graphs. Some graphs are only distinct points.
They are called discrete graphs
The graph on theme park
attendance is an example of a
discrete graph. It consists of
distinct points because each
year is distinct and people are
counted in whole numbers only.
The values between whole
numbers are not included, since
they have no meaning for the
situation.
Holt Algebra 1
4-2 Relations and Functions
Example 2A: Sketching Graphs for Situations
Sketch a graph for the situation. Tell whether
the graph is continuous or discrete.
•
•
•
•
•
initially increases
remains constant
decreases to a stop
increases
remains constant
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Speed
A truck driver enters a street, drives at a
constant speed, stops at a light, and then
continues.
As time passes during the trip
y
(moving left to right along the
x-axis) the truck's speed (y-axis)
does the following:
Time
x
The graph is continuous.
4-2 Relations and Functions
Example 2B: Sketching Graphs for Situations
Sketch a graph for the situation. Tell whether
the graph is continuous or discrete.
A small bookstore sold between 5 and 8
books each day for 7 days.
The number of books sold
(y-axis) varies for each day
(x-axis).
Since the bookstore accounts
for the number of books sold
at the end of each day, the
graph is 7 distinct points.
The graph is discrete.
Holt Algebra 1
4-2 Relations and Functions
Check It Out! Example 2a
Sketch a graph for the situation. Tell whether
the graph is continuous or discrete.
Jamie is taking an 8-week keyboarding class.
At the end of each week, she takes a test to
find the number of words she can type per
minute. She improves each week.
Each week (x-axis) her typing
speed is measured. She gets
a separate score (y-axis) for
each test.
Since each score is separate, the
graph consists of distinct units.
The graph is discrete.
Holt Algebra 1
4-2 Relations and Functions
Example 3: Writing Situations for Graphs
Write a possible situation for the given graph.
Step 1 Identify labels.
x-axis: time y-axis: speed
Step 2 Analyze sections.
over time, the speed:
• initially decreases,
• remains constant,
• and then decreases to zero.
Possible Situation:
A car approaching traffic slows down, drives at a
constant speed, and then slows down until coming
to a complete stop.
Holt Algebra 1
4-2 Relations and Functions
Check It Out! Example 2b
Sketch a graph for the situation. Tell whether
the graph is continuous or discrete.
Henry begins to drain a water tank by opening
a valve. Then he opens another valve. Then he
closes the first valve. He leaves the second
valve open until the tank is empty.
Water Level
Water tank
As time passes while draining the
tank (moving left to right along the
x-axis) the water level (y-axis) does
the following:
• initially declines
• decline more rapidly
• and then the decline slows down.
Time
Holt Algebra 1
The graph is continuous.
4-2 Relations and Functions
Check It Out! Example 3
Write a possible situation for the given graph
Step 1 Identify labels.
x-axis: students y-axis: pizzas
Step 2 Analyze sections.
As students increase, the
pizzas do the following:
• initially remains constant,
• and then increases to a new
constant.
Possible Situation:
When the number of students reaches a certain
point, the number of pizzas bought increases.
Holt Algebra 1
4-2 Relations and Functions
Lesson Quiz: Part I
1. Express the relation {(–2, 5), (–1, 4), (1, 3),
(2, 4)} as a table, as a graph, and as a
mapping diagram.
2. Give the domain and range of the relation.
Holt Algebra 1
4-2 Relations and Functions
Lesson Quiz Contd.
3. Give the domain and range of the
relation. Tell whether the relation is a
function. Explain.
Holt Algebra 1
4-2 Relations and Functions
CW/HW for today:
Pages 239-242
#2-20(E), #32-34 (E)
Page 243 # 1 – 3 (all)
Page 233-235
#4 – 16(E), # 23 – 24(both)
Holt Algebra 1