2.1
Before
Now
Why?
Key Vocabulary
• relation
• domain
• range
• function
• equation in two
variables
• linear function
Represent Relations
and Functions
You solved linear equations.
You will represent relations and graph linear functions.
So you can model changes in elevation, as in Ex. 48.
A relation is a mapping, or pairing, of input values with output values. The set of
input values is the domain, and the set of output values is the range.
For Your Notebook
KEY CONCEPT
Representing Relations
A relation can be represented in the following ways.
Ordered Pairs
Table
(22, 2)
x
(22, 22)
(0, 1)
2
22
22
0
1
3
1
Mapping Diagram
y
y
22
(3, 1)
EXAMPLE 1
Graph
Input
Output
22
22
0
2
3
1
1
1
x
Represent relations
Consider the relation given by the ordered pairs (22, 23), (21, 1), (1, 3), (2, 22),
and (3, 1).
a. Identify the domain and range.
b. Represent the relation using a graph and a mapping diagram.
Solution
a. The domain consists of all the x-coordinates: 22, 21, 1, 2, and 3.
The range consists of all the y-coordinates: 23, 22, 1, and 3.
REVIEW GRAPHING
For help with plotting
points in a coordinate
plane, see p. 987.
b. Graph
Mapping Diagram
y
1
1
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x
Input
Output
22
21
1
2
3
23
22
1
3
Chapter 2 Linear Equations and Functions
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FUNCTIONS A function is a relation for which each input has exactly one output.
If any input of a relation has more than one output, the relation is not a function.
EXAMPLE 2
Identify functions
Tell whether the relation is a function. Explain.
AVOID ERRORS
A relation can map
more than one input
onto the same output
and still be a function.
a.
Output
2
22
24
3
1
24
4
Output
23
21
2
4
b.
Input
Input
21
2
3
Solution
a. The relation is a function because each input is mapped onto exactly
one output.
b. The relation is not a function because the input 1 is mapped onto both
21 and 2.
"MHFCSB
✓
at classzone.com
GUIDED PRACTICE
for Examples 1 and 2
1. Consider the relation given by the ordered pairs (24, 3), (22, 1), (0, 3), (1, 22),
and (22, 24).
a. Identify the domain and range.
b. Represent the relation using a table and a mapping diagram.
2. Tell whether the relation
is a function. Explain.
x
22
21
0
1
3
y
24
24
24
24
24
VERTICAL LINE TEST You can use the graph of a relation to determine whether it
is a function by applying the vertical line test.
For Your Notebook
KEY CONCEPT
REVIEW LOGICAL
STATEMENTS
For help with “if and
only if” statements, see
p. 1002.
Vertical Line Test
A relation is a function if and only if no vertical line intersects the graph of the
relation at more than one point.
Function
Not a function
y
1
y
1
1
x
1
x
2.1 Represent Relations and Functions
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EXAMPLE 3
Use the vertical line test
BASKETBALL The first graph below plots average points per game versus age at
the end of the 2003–2004 NBA regular season for the 8 members of the Minnesota
Timberwolves with the highest averages. The second graph plots average
points per game versus age for one team member, Kevin Garnett, over his
first 9 seasons. Are the relations shown by the graphs functions? Explain.
READING GRAPHS
The zigzag symbol on
the horizontal axis of
each graph indicates
that values of x were
skipped.
✓
Kevin Garnett
Average points
Average points
Timberwolves
y
30
20
10
0
0 26
28
30
32
Age (years)
34 x
y
30
20
10
0
0 20
22
24
26
Age (years)
28 x
Solution
The team graph does not represent a function because vertical lines at x 5 28
and x 5 29 each intersect the graph at more than one point. The graph for Kevin
Garnett does represent a function because no vertical line intersects the graph at
more than one point.
GUIDED PRACTICE
for Example 3
3. WHAT IF? In Example 3, suppose that Kevin Garnett averages 24.2 points
per game in his tenth season as he did in his ninth. If the relation given by
the second graph is revised to include the tenth season, is the relation still a
function? Explain.
EQUATIONS IN TWO VARIABLES Many functions can be described by an equation
in two variables, such as y 5 3x 2 5. The input variable (in this case, x) is called
the independent variable. The output variable (in this case, y) is called the
dependent variable because its value depends on the value of the input variable.
An ordered pair (x, y) is a solution of an equation in two variables if substituting
x and y in the equation produces a true statement. For example, (2, 1) is a solution
of y 5 3x 2 5 because 1 5 3(2) 2 5 is true. The graph of an equation in two
variables is the set of all points (x, y) that represent solutions of the equation.
KEY CONCEPT
For Your Notebook
Graphing Equations in Two Variables
To graph an equation in two variables, follow these steps:
STEP 1
Construct a table of values.
STEP 2 Plot enough points from the table to recognize a pattern.
STEP 3 Connect the points with a line or a curve.
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Chapter 2 Linear Equations and Functions
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EXAMPLE 4
Graph an equation in two variables
Graph the equation y 5 22x 2 1.
Solution
y
Construct a table of values.
STEP 1
x
22
21
0
1
2
y
3
1
21
23
25
1
1
x
STEP 2 Plot the points. Notice that they all lie on a line.
STEP 3 Connect the points with a line.
LINEAR FUNCTIONS The function y 5 22x 2 1 in Example 4 is a linear function
because it can be written in the form y 5 mx 1 b where m and b are constants.
The graph of a linear function is a line. By renaming y as f(x), you can write
y 5 mx 1 b using function notation.
READING
The parentheses in
f(x) do not indicate
multiplication. The
symbol f(x) does not
mean “f times x.”
y 5 mx 1 b
Linear function in x-y notation
f(x) 5 mx 1 b
Linear function in function notation
The notation f(x) is read “the value of f at x,” or simply “f of x,” and identifies x as
the independent variable. The domain consists of all values of x for which f (x) is
defined. The range consists of all values of f(x) where x is in the domain of f .
EXAMPLE 5
Classify and evaluate functions
Tell whether the function is linear. Then evaluate the function when x 5 24.
a. f(x) 5 2x 2 2 2x 1 7
b. g(x) 5 5x 1 8
Solution
a. The function f is not linear because it has an x2-term.
f(x) 5 2x 2 2 2x 1 7
Write function.
f(24) 5 2(24)2 2 2(24) 1 7
Substitute 24 for x.
Simplify.
5 21
b. The function g is linear because it has the form g(x) 5 mx 1 b.
REPRESENT
FUNCTIONS
Letters other than f,
such as g or h, can also
name functions.
✓
g(x) 5 5x 1 8
g(24) 5 5(24) 1 8
5 212
GUIDED PRACTICE
Write function.
Substitute 24 for x.
Simplify.
for Examples 4 and 5
4. Graph the equation y 5 3x 2 2.
Tell whether the function is linear. Then evaluate the function when x 5 22.
5. f(x) 5 x 2 1 2 x 3
6. g(x) 5 24 2 2x
2.1 Represent Relations and Functions
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DOMAINS IN REAL LIFE In Example 5, the domain of each function is all real
numbers because there is an output for every real number x. In real life, you may
need to restrict the domain so that it is reasonable in the given situation.
EXAMPLE 6
Use a function in real life
DIVING A diver using a Diver Propulsion Vehicle (DPV)
descends to a depth of 130 feet. The pressure P (in
atmospheres) on the diver is given by P(d) 5 1 1 0.03d
where d is the depth (in feet). Graph the function, and
determine a reasonable domain and range. What is the
pressure on the diver at a depth of 33 feet?
Solution
Pressure on a Diver
The graph of P(d) is shown. Because the depth
varies from 0 feet to 130 feet, a reasonable
domain is 0 ≤ d ≤ 130.
Pressure
(atmospheres)
P(d)
The minimum value of P(d) is P(0) 5 1, and the
maximum value of P(d) is P(130) 5 4.9. So, a
reasonable range is 1 ≤ P(d) ≤ 4.9.
c At a depth of 33 feet, the pressure on the diver
is P(33) 5 1 1 0.03(33) < 2 atmospheres, which
you can verify from the graph.
✓
GUIDED PRACTICE
4
2
0
(33, 2)
0
40 80 120 d
Depth (ft)
for Example 6
7. OCEAN EXPLORATION In 1960, the deep-sea vessel Trieste descended to an
estimated depth of 35,800 feet. Determine a reasonable domain and range of
the function P(d) in Example 6 for this trip.
2.1
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 7, 17, and 45
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 9, 20, 24, 40, 46, and 49
SKILL PRACTICE
1. VOCABULARY Copy and complete: In the equation y 5 x 1 5, x is the ?
variable and y is the ? variable.
2. ★ WRITING Describe how to find the domain and range of a relation given by
a set of ordered pairs.
EXAMPLE 1
on p. 72
for Exs. 3–9
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REPRESENTING RELATIONS Identify the domain and range of the given relation.
Then represent the relation using a graph and a mapping diagram.
3. (22, 3), (1, 2), (3, 21), (24, 23)
4. (5, 22), (23, 22), (3, 3), (21, 21)
5. (6, 21), (22, 23), (1, 8), (22, 5)
6. (27, 4), (2, 25), (1, 22), (23, 6)
7. (5, 20), (10, 20), (15, 30), (20, 30)
8. (4, 22), (4, 2), (16, 24), (16, 4)
Chapter 2 Linear Equations and Functions
10/20/05 8:55:45 AM
9. ★ MULTIPLE CHOICE What is the domain of the relation given by the ordered pairs
(24, 2), (21, 23), (1, 4), (1, 23), and (2, 1)?
A 23, 1, 2, and 4
B 24, 21, 1, and 2
C 24, 23, 21, and 2
D 24, 23, 21, 1, 2, and 4
EXAMPLE 2
IDENTIFYING FUNCTIONS Tell whether the relation is a function. Explain.
on p. 73
for Exs. 10–20
10.
11.
Input
Output
5
25
22
24
2
23
21
1
1
2
21
2
Input
Output
26
12.
Input
13.
Output
Input
4
28
21
2
24
5
21
0
23
4
1
4
Output
0
ERROR ANALYSIS Describe and correct the error in the student’s work.
14.
15.
The relation given by the ordered
pairs (24, 2), (21, 5), (3, 6), and
(7, 2) is not a function because
the inputs 24 and 7 are both
mapped to the output 2.
x
0
1
2
1
0
y
5
6
7
8
9
The relation given by the table is
a function because there is only
one value of x for each value of y.
IDENTIFYING FUNCTIONS Tell whether the relation is a function. Explain.
16. (3, 22), (0, 1), (1, 0), (22, 21), (2, 21)
17. (2, 25), (22, 5), (21, 4), (22, 0), (3, 24)
18. (0, 1), (1, 0), (2, 3), (3, 2), (4, 4)
19. (21, 21), (2, 5), (4, 8), (25, 29), (21, 25)
20. ★ MULTIPLE CHOICE The relation given by the ordered pairs (26, 3), (22, 4),
(1, 5), and (4, 0) is a function. Which ordered pair can be included with this
relation to form a new relation that is also a function?
A (1, 25)
B (6, 3)
C (22, 19)
D (4, 4)
EXAMPLE 3
VERTICAL LINE TEST Use the vertical line test to tell whether the relation is
on p. 74
for Exs. 21–23
a function.
21.
22.
y
23.
y
y
2
1
1
x
1
2
x
1
x
24. ★ SHORT RESPONSE Explain why a relation is not a function if a vertical line
intersects the graph of the relation more than once.
EXAMPLE 4
GRAPHING EQUATIONS Graph the equation.
on p. 75
for Exs. 25–33
25. y 5 x 1 2
26. y 5 2x 1 5
27. y 5 3x 1 1
28. y 5 5x 2 3
29. y 5 2x 2 7
30. y 5 23x 1 2
31. y 5 22x
1x 1 2
32. y 5 }
2
3x 2 1
33. y 5 2}
4
2.1 Represent Relations and Functions
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EXAMPLE 5
on p. 75
for Exs. 34–39
EVALUATING FUNCTIONS Tell whether the function is linear. Then evaluate the
function for the given value of x.
34. f(x) 5 x 1 15; f (8)
35. f(x) 5 x 2 1 1; f(23)
36. f(x) 5 x 1 10; f(24)
37. f(x) 5 6; f(2)
38. g(x) 5 x 3 2 2x2 1 5x 2 8; g(25)
2 x; h(15)
39. h(x) 5 7 2 }
3
40. ★ SHORT RESPONSE Which, if any, of the relations described by the
equations y 5 x, x 5 y, and y 5 x represent functions? Explain.
41. CHALLENGE Let f be a function such that f(a 1 b) 5 f (a) 1 f(b) for all real
numbers a and b. Show that f(2a) 5 2 p f (a) and that f (0) 5 0.
PROBLEM SOLVING
42. BICYCLING The graph shows the ages of the top
three finishers in the Mt. Washington Auto Road
Bicycle Hillclimb each year from 2002 through
2004. Do the ordered pairs (age, finishing place)
represent a function? Explain.
on p. 74
for Exs. 42–43
Finishing
place
EXAMPLE 3
y
3
2
1
0
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
games started and the number of wins for each
starting pitcher on a baseball team during a
regular season. Do the ordered pairs (starts, wins)
represent a function? Explain.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
Wins
43. BASEBALL The graph shows the number of
20
22
24
Age (years)
26 x
y
10
5
0
0 15 18 21 24 27 30 33 x
Starts
44.
GEOMETRY The volume V of a cube with edge length s is given by the
function V(s) 5 s3. Find V(4). Explain what V(4) represents.
45.
GEOMETRY The volume V of a sphere with radius r is given by the
4 πr 3. Find V(6). Explain what V(6) represents.
function V(r) 5 }
3
EXAMPLE 6
on p. 76
for Exs. 46–48
46. ★ SHORT RESPONSE For the period 1974–2004, the average price p (in
dollars) of a theater ticket in the United States can be modeled by the
function p(t) 5 0.144t 1 1.89 where t is the number of years since 1974.
Determine a reasonable domain and range for p(t). Explain the meaning
of the range.
47. MULTI-STEP PROBLEM Anthropologists can estimate a person’s height
from the length of certain bones. The height h (in inches) of an adult
human female can be modeled by the function h(l) 5 1.95l 1 28.7 where
l is the length (in inches) of the femur, or thigh bone. The function is valid
for femur lengths between 15 inches and 24 inches, inclusive.
a. Graph the function, and determine a reasonable domain and range.
b. Suppose a female’s femur is 15.5 inches long. About how tall was she?
c. If an anthropologist estimates a female’s height as 5 feet 11 inches,
about how long is her femur?
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5 WORKED-OUT SOLUTIONS
Chapter 2 Linear
and Functions
on p. Equations
WS1
★
5 STANDARDIZED
TEST PRACTICE
10/20/05 8:55:48 AM
48. MOUNTAIN CLIMBING A climber on Mount
Rainier in Washington hikes from an elevation
of 5400 feet above sea level to Camp Muir,
which has an elevation of 10,100 feet. The
elevation h (in feet) as the climber ascends
can be modeled by h(t) 5 1000t 1 5400 where
t is the time (in hours). Graph the function,
and determine a reasonable domain and range.
What is the climber’s elevation after hiking
3.5 hours?
49. ★ EXTENDED RESPONSE The table shows
the populations of several states and their
electoral votes in the 2004 and 2008 U.S.
presidential elections. The figures are based
on U.S. census data for the year 2000.
Population
(millions), p
Electoral
votes, v
California
33.87
55
Florida
15.98
27
Illinois
12.42
21
New York
18.98
31
Ohio
11.35
20
Pennsylvania
12.28
21
Texas
20.85
34
State
a. Identify the domain and range of the
relation given by the ordered pairs (p, v).
b. Is the relation from part (a) a function?
Explain.
c. Is the relation given by the ordered pairs
(v, p) a function? Explain.
50. CHALLENGE The table shows ground
shipping charges for an online retail store.
a. Is the shipping cost a function of the
merchandise cost? Explain.
b. Is the merchandise cost a function of the
shipping cost? Explain.
Merchandise cost
Shipping cost
$.01–$30.00
$4.50
$30.01–$60.00
$7.25
$60.01–$100.00
$9.50
Over $100.00
$12.50
MIXED REVIEW
PREVIEW
Evaluate the expression for the given values of x and y. (p. 10)
Prepare for
Lesson 2.2
in Exs. 51–54.
y23
51. } when x 5 6 and y 5 2
x24
y28
52. } when x 5 3 and y 5 4
x22
y 2 (25)
53. } when x 5 23 and y 5 23
x21
24 2 y
54. } when x 5 217 and y 5 8
15 2 x
Solve the equation. Check your solution. (p. 18)
55. 3x 1 16 5 31
56. 24x 2 7 5 17
57. 5x 1 12 5 23x 2 4
58. 5 2 8z 5 25 1 4z
5 (3v 2 4) 5 30
59. }
2
60. 6(4w 1 1) 5 1.5(8w 1 18)
Solve the inequality. Then graph the solution. (p. 41)
61. 2x 2 6 > 8
1x 1 7 > 0
62. }
4
63. 15 2 2x ≤ 7
64. 4 2 x < 3
65. 27 < 6x 2 1 < 5
66. x 2 2 ≤ 1 or 4x 1 3 ≥ 19
EXTRA PRACTICE for Lesson 2.1, p. 1011
n2pe-0201.indd 79
ONLINE
QUIZ
at classzone.com
2.1 Represent
Relations
and Functions
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