3.4
a) Write the equation of the line through (8.24, 1015.5) and
(7.26, 717.1) and express w as a linear function of d.
b) What is the flow when the depth is 7.81 feet?
c) Is the flow increasing or decreasing as the depth
increases?
a) w 304.5d 1493.5 b) 884.6 ft3/sec
c) increasing
90. Buying stock. On July 2, 1998 a mutual fund manager
spent $5,031,250 on x shares of Ford Motor Stock at $58.25
per share and y shares of General Motors stock at $47.50
per share.
a) Write a linear equation that models this situation.
b) If 35,000 shares of Ford were purchased, then how
many shares of GM were purchased?
c) What are the intercepts of the graph of the linear equation? Interpret the intercepts.
d) As the number of shares of Ford increases, does the
number of shares of GM increase or decrease?
a) 58.25x 47.50y 5,031,250 b) 63,000
c) (0, 105,921.1), (86,373.4, 0), The intercepts give the
number of shares if all of the money was spent on
only one type of stock. d) decrease
GM shares (in thousands)
150
100
50
0
0
50
100
Ford shares (in thousands)
FIGURE FOR EXERCISE 90
3.4
In this
section
●
Definition
●
Graphing Linear Inequalities
●
The Test Point Method
●
Graphing Compound
Inequalities
●
Applications
Linear Inequalities and Their Graphs
(3-31)
151
GET TING MORE INVOLVED
91. Exploration. Plot the points (1, 1), (2, 3), (3, 4), (4, 6), and
(5, 7) on graph paper. Use a ruler to draw a straight line that
“best fits” the five points. The line drawn does not necessarily have to go through any of the five points.
a) Estimate the slope and y-intercept for the line drawn and
write an equation for the line in slope-intercept form.
b) For each x-coordinate from 1 through 5, find the
difference between the given y-coordinate and the ycoordinate on your line.
c) To determine how well you have done, square each
difference that you found in part (b) and then find the
sum of those squares. Compare your sum with your
classmates’ sums. The person with the smallest sum has
done the best job of fitting a line to the five given points.
G R A P H I N G C ALC U L ATO R
EXERCISES
92. Graph the equation y 0.5x 1 using the standard viewing window. Adjust the range of y-values so that the line
goes from the lower left corner of your viewing window to
the upper right corner.
93. Graph y x 3000, using a viewing window that shows
both the x-intercept and the y-intercept.
94. Graph y 2x 400 and y 0.5x 1 on the same
screen, using the viewing window 500 x 500 and
1000 y 1000. Should these lines be perpendicular?
Explain.
The lines are perpendicular and will appear so in a window
in which the length of one unit on the x-axis is equal to the
length of one unit on the y-axis.
95. The lines y 2x 3 and y 1.9x 2 are not parallel.
Find a viewing window in which the lines intersect.
Estimate the point of intersection.
The lines intersect at (50, 97).
LINEAR INEQUALITIES AND
THEIR GRAPHS
In the first three sections of this chapter you studied linear equations. We now turn
our attention to linear inequalities.
Definition
A linear inequality is a linear equation with the equal sign replaced by an inequality
symbol.
Linear Inequality
If A, B, and C are real numbers with A and B not both zero, then
Ax By C
is called a linear inequality. In place of , we can also use , , or .
152
(3-32)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
Graphing Linear Inequalities
study
tip
Working problems one hour
per day every day of the week
is better than working problems for 7 hours on one day of
the week. It is usually better to
spread out your study time
than to try and learn everything in one big session.
Consider the inequality x y 1. If we solve the inequality for y, we get
y x 1.
Which points in the xy-plane satisfy this inequality? We want the points where the
y-coordinate is larger than the x-coordinate plus 1. If we locate a point on the
line y x 1, say (2, 3), then the y-coordinate is equal to the x-coordinate plus 1.
If we move upward from that point, to say (2, 4), the y-coordinate is larger than the
x-coordinate plus 1. Because this argument can be made at every point on the line, all
points above the line satisfy y x 1. Likewise, points below the line satisfy
y x 1. The solution sets, or graphs, for the inequality y x 1 and the
inequality y x 1 are the shaded regions shown in Figs. 3.22(a) and 3.22(b). In
each case the line y x 1 is dashed to indicate that points on the line do not
satisfy the inequality and so are not in the solution set. If the inequality symbol
is or , then points on the boundary line also satisfy the inequality, and the line is
drawn solid.
y
y>x+1
y
3
2
y=x+1
3
2
y<x+1
y=x+1
–3 –2
1
–1
–2
–3
2
3
x
–3 –2
–1
–2
–3
1
2
3
x
(b)
(a)
FIGURE 3.22
Every nonvertical line divides the xy-plane into two regions. One region is
above the line, and the other is below the line. A vertical line also divides the plane
into two regions, but one is on the left side of the line and the other is on the right
side of the line. An inequality involving only x has a vertical boundary line, and its
graph is one of those regions.
Graphing a Linear Inequality
1. Solve the inequality for y, then graph y mx b.
y mx b is satisfied above the line.
y mx b is satisfied on the line itself.
y mx b is satisfied below the line.
2. If the inequality involves x and not y, then graph the vertical line x k.
x k is satisfied to the right of the line.
x k is satisfied on the line itself.
x k is satisfied to the left of the line.
E X A M P L E
1
Graphing linear inequalities
Graph each inequality.
1
b) y 2x 1
a) y x 1
2
c) 3x 2y 6
3.4
helpful
(3-33)
Linear Inequalities and Their Graphs
153
Solution
a) The set of points satisfying this inequality is the region below the line y 1x 1.
2
To show this region, we first graph the boundary line y 1x 1. The slope of
2
the line is 1, and the y-intercept is (0, 1). Start at (0, 1) on the y-axis, then
2
rise 1 and run 2 to get a second point of the line. We draw the line dashed because
points on the line do not satisfy this inequality. The solution set to the inequality
is the shaded region shown in Fig. 3.23.
b) Because the inequality symbol is , every point on or above the line satisfies
this inequality. To show that the line y 2x 1 is included, we make it a
solid line. See Fig. 3.24.
c) First solve for y:
hint
Why do we keep drawing
graphs? When we solve
2x 1 7, we don’t bother
to draw a graph showing 3
because the solution set is so
simple. However, the solution
set to a linear inequality is a
very large set of ordered pairs.
Graphing gives us a way to
visualize the solution set.
3x 2y 6
2y 3x 6
3
y x 3 Divide by 2 and reverse the inequality.
2
To graph this inequality, use a dashed line for the boundary y 32 x 3 and shade
the region above the line. See Fig. 3.25 for the graph.
y
y
5
4
3
y
3
2
1
y>
1
–4 –3 – 2 – 1
–2
–3
y ≥ –2x + 1
2
y<
3
1
—
x
2
4
–1
x
–3 –2 –1
–1
–2
–3
1 2
3
4
5
x
3
—
x
2
4
3
–3
2
1
–3 –2 –1
–1
–2
–3
–4
–5
–4
–5
FIGURE 3.23
FIGURE 3.24
1
3
4
5
x
–5
FIGURE 3.25
■
CAUTION
In Example 1(c) we solved the inequality for y before graphing the line. We did that because corresponds to the region below the line and
corresponds to the region above the line only when the inequality is solved for y.
E X A M P L E
2
Inequalities with horizontal and vertical boundaries
Graph the inequalities.
a) y 5
b) x 4
Solution
a) The line y 5 is the horizontal line with y-intercept (0, 5). Draw a solid horizontal line and shade below it as in Fig. 3.26 on the next page.
b) The points that satisfy x 4 lie to the right of the vertical line x 4. The
solution set is shown in Fig. 3.27 on the next page.
154
(3-34)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
y
y
4
3
2
1
5
4
3
2
1
y≤5
–4 –3 –2 –1
–1
–2
–3
1
2
3
4
x
–4
–3 –2 –1
–1
–2
–3
–4
FIGURE 3.26
x>4
1
2
3
x
5
■
FIGURE 3.27
The Test Point Method
The graph of any line Ax By C separates the xy-plane into two regions. Every
point on one side of the line satisfies the inequality Ax By C, and every point
on the other side satisfies the inequality Ax By C. We can use these facts to
graph an inequality by the test point method:
1. Graph the corresponding equation.
2. Choose any point not on the line.
3. Test to see whether the point satisfies the inequality.
If the point satisfies the inequality, then the solution set is the region containing the
test point. If not, then the solution set is the other region. With this method, it is not
necessary to solve the inequality for y.
E X A M P L E
study
3
tip
Students who have difficulty
with algebra often schedule a
class that meets one day per
week so that they do not have
to see it as often. However,
students usually do better in
classes that meet more often
for shorter time periods.
Using the test point method
Graph the inequality 3x 4y 7.
Solution
First graph the equation 3x 4y 7 using the x-intercept and the y-intercept. If
x 0, then y 7. If y 0, then x 7. Use the x-intercept 7, 0 and the y4
3
3
intercept 0, 7 to graph the line as shown in Fig. 3.28(a). Select a point on one
4
Test point
y
y
4
3
2
4
3
2
1
–5 –4 –3 –2 –1
–1
3x – 4y = 7
1
3
4
5
x
–5 –4 –3 –2 –1
1
–3
–3
–4
–5
–4
–5
(a)
(b)
FIGURE 3.28
3
4
3x – 4y > 7
5
x
3.4
(3-35)
Linear Inequalities and Their Graphs
155
side of the line, say (0, 1), to test in the inequality. Because
3(0) 4(1) 7
is false, the region on the other side of the line satisfies the inequality. The graph of
■
3x 4y 7 is shown in Fig. 3.28(b).
Graphing Compound Inequalities
We can write compound inequalities with two variables just as we do for one variable. For example,
1
yx3
and
y x 2
2
is a compound inequality. Because the inequalities are connected by the word and,
a point is in the solution set to the compound inequality if and only if it is in the
solution sets to both of the individual inequalities. So the graph of this compound
inequality is the intersection of the solution sets to the individual inequalities.
E X A M P L E
4
Graphing a compound inequality with and
1
Graph the compound inequality y x 3 and y 2 x 2.
Solution
1
We first graph the equations y x 3 and y 2 x 2. These lines divide the
plane into four regions as shown in Fig. 3.29(a). Now test one point of each region
to determine which region satisfies the compound inequality. Test the points (3, 3),
(0, 0), (4, 5), and (5, 0):
1
333
and
3 2 3 2
003
and
0 2 0 2
5 4 3
and
5 2 4 2
053
and
5 2 0 2
1
y=–—
x+2
2
1
1
Both inequalities are correct.
First inequality is incorrect.
Both inequalities are incorrect.
y
y
5
4
3
5
4
1
1
–5 –4 –3 –2 –1
–1
–2
–3
y=x–3
1
Second inequality is incorrect.
3
1
2
3 4
Test points
x
–5 –4 –3 –2 –1
–1
–2
–3
1
2
3
x
–4
–5
–4
–5
(a)
(b)
FIGURE 3.29
The only point that satisfies both inequalities is (0, 0). So the solution set to the compound inequality consists of all points in the region containing (0, 0). The graph of
■
the compound inequality is shown in Fig. 3.29(b).
156
(3-36)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
Compound inequalities are also formed by connecting individual inequalities
with the word or. A point satisfies a compound inequality connected by or if and
only if it satisfies one or the other or both of the individual inequalities. The graph
is the union of the graphs of the individual inequalities.
E X A M P L E
5
Graphing a compound inequality with or
Graph the compound inequality
2x 3y 6
helpful
hint
When graphing a compound
inequality connected with“or,”
shade the region that satisfies
the first inequality and then
shade the region that satisfies
the second inequality. If the
inequalities are connected
with “and,” then you must be
careful not to shade too much.
or
x 2y 4.
Solution
First graph the lines 2x 3y 6 and x 2y 4. If we graph the lines using
x- and y-intercepts, then we do not have to solve the equations for y. The lines are
shown in Fig. 3.30(a). The graph of the compound inequality is the set of all points
that satisfy either one inequality or the other (or both). Test the points (0, 0), (3, 2),
(0, 5), and (3, 2). You should verify that only (0, 0) fails to satisfy at least one of
the inequalities. So only the region containing the origin is left unshaded. The graph
of the compound inequality is shown in Fig. 3.30(b).
y
5
4
y
2x – 3y = – 6
5
4
Test points
3
3
1
–5
–3 –2 –1
–1
–2
–3
1
1
2
3
x
4
–5
x + 2y = 4
–3 –2 –1
–1
–2
–3
(a)
1
2
3
4
x
(b)
FIGURE 3.30
■
In the next example we graph absolute value inequalities by writing equivalent
compound inequalities.
E X A M P L E
6
Graphing absolute value inequalities
Graph each absolute value inequality.
a) y 2x 3
b) x y 1
Solution
a) The inequality y 2x 3 is equivalent to 3 y 2x 3, which is
equivalent to the compound inequality
y 2x 3
and
y 2x 3.
First graph the lines y 2x 3 and y 2x 3 as shown in Fig. 3.31(a) on
the next page. These lines divide the plane into three regions. Test a point from
each region in the original inequality, say (5, 0), (0, 1), and (5, 0):
0 2(5) 3
1203
0253
10 3
13
10 3
3.4
helpful
hint
Remember that absolute
value of a quantity is its distance from 0 (Section 2.6). If
w 3, then w is less than 3
units from 0:
3 w 3
If w 1, then w is more than
1 unit away from 0:
w1
or w 1
y
y
5
4
3
5
4
3
y – 2x  ≤ 3
y – 2x = 3
y – 2x = – 3
1
–5 – 4 – 3 – 2 – 1
–1
–2
–3
In Example 6 we are using an
expression in place of w.
1
2
3
(3-37)
Linear Inequalities and Their Graphs
4 5
157
1
x
Test points
–5 –4 –3 –2 –1
–1
–2
–3
–5
1
2
3
4 5
x
–5
(a)
(b)
FIGURE 3.31
Only (0, 1) satisfies the original inequality. So the region satisfying the absolute
value inequality is the shaded region containing (0, 1) as shown in Fig. 3.31(b).
The boundary lines are solid because of the symbol.
b) The inequality x y 1 is equivalent to
x y 1 or x y 1.
First graph the lines x y 1 and x y 1 as shown in Fig. 3.32(a). Test a
point from each region in the original inequality, say (4, 0), (0, 0), and (4, 0):
4 0 1
001
401
41
01
41
y
5
y
–5 –4 –3 –2
–1
–2
–3
5
x – y = –1
4
3
2
1
x – y  > 1
4
3
2
x–y=1
1
2
3
4
5
x
–5 –4 –3 –2
Test points
–1
1 2
3
4
5
x
–2
–3
–4
–5
–4
–5
(a)
(b)
FIGURE 3.32
Because (4, 0) and (4, 0) satisfy the inequality, we shade those regions as
shown in Fig. 3.32(b). The boundary lines are dashed because of the symbol.
■
Applications
In real situations x and y often represent quantities or amounts, which cannot be
negative. In this case our graphs are restricted to the first quadrant, where x and y
are both nonnegative.
158
(3-38)
Chapter 3
E X A M P L E
7
Graphs and Functions in the Cartesian Coordinate System
Inequalities in business
The manager of a furniture store can spend a maximum of $3000 on advertising per
week. It costs $50 to run a 30-second ad on an AM radio station and $75 to run the
ad on an FM station. Graph the region that shows the possible numbers of AM and
FM ads that can be purchased and identify some possibilities.
Solution
If x represents the number of AM ads and y represents the number of FM ads, then
x and y must satisfy the inequality 50x 75y 3000. Because the number of ads
cannot be negative, we also have x 0 and y 0. So we graph only points in the
y
60
Number of FM ads
50
40
30
20
10
x
0
10
20 30 40 50
Number of AM ads
60
FIGURE 3.33
M A T H
A T
W O R K
“We will return after these messages.” We often hear
these words on television and radio just before several
minutes of commercials. Carolanne Johnson, Account
Executive and Media Salesperson for WBOQ, a classical
radio station, is involved in every step of creating such
advertisements.
MEDIA
The first step is finding clients that are consistent with
SALESPERSON
the station’s image. Ms. Johnson generates her own leads
from a number of sources, such as print ads and billboards. The next steps are sitting
down with the client, gathering information about the product or service, assessing the
competition, and finally determining how much of the client’s advertising budget
should be spent on radio. Typically, this can be 2% to 4% of the total budget.
Radio ads usually run for 60 seconds, but reminder ads can be as short as 30 seconds. Some of the radio spots are time-sensitive and run 40 to 60 times a month for
a specific month. Other clients are concerned with image building and may sponsor
one particular broadcast every day for the whole year.
Ms. Johnson is concerned that the clients receive an adequate return on their
investment. She is constantly reviewing the budget and making sure that the commercials present what the client wishes to project.
Example 7 and Exercise 73 of this section give problems that involve allocation
of advertising dollars.
3.4
Linear Inequalities and Their Graphs
(3-39)
159
first quadrant that satisfy 50x 75y 3000. The line 50x 75y 3000 goes
through (0, 40) and (60, 0). The inequality is satisfied below this line. The region
showing the possible numbers of AM ads and FM ads is shown in Fig. 3.33. We
shade the entire region in Fig. 3.33, but only points in the shaded region in which
both coordinates are whole numbers actually satisfy the given condition. For example, 40 AM ads and 10 FM ads could be purchased. Other possibilities are 30 AM
ads and 20 FM ads, or 10 AM ads and 10 FM ads.
WARM-UPS
True or false? Explain your answer.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
3. 4
The point (2, 3) satisfies the inequality y 3x 2. True
The graph of 3x y 2 is the region above the line 3x y 2. False
The graph of 3x y 5 is the region below the line y 3x 5. True
The graph of x 3 is the region to the left of the vertical line x 3. False
The graph of y x 3 and y 2x 6 is the intersection of two regions.
True
The graph of y 2x 3 or y 3x 5 is the union of two regions. True
The ordered pair (2, 5) satisfies y 3x 5 and y 2x 3. False
The ordered pair (3, 2) satisfies y 3x 6 or y x 5. True
The inequality 2x y 4 is equivalent to 2x y 4 and 2x y 4.
False
The inequality x y 3 is equivalent to x y 3 or x y 3.
True
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a linear inequality?
Alinear inequality is an inequality of the form Ax By C
(or using , , or ), where A, B, and C are real numbers
and A and B are not both zero.
2. How do we usually illustrate the solution set to a linear
inequality in two variables.
The solution set to a linear inequality in two variables is
usually illustrated with a graph.
3. How do you know whether the line should be solid or
dashed when graphing a linear inequality?
If the inequality includes equality, then the line should be
solid.
4. How do you know which side of the line to shade when
graphing a linear inequality?
We shade the side on which the inequality is satisfied.
5. What is the test point method used for?
The test point method is used to determine which side of
the boundary line to shade.
6. How do you graph a compound inequality?
To graph a compound inequality, we find either the union or
intersection of the regions determined by each simple
inequality.
Graph each linear inequality. See Examples 1 and 2.
7. y x 2
9. y 2x 1
8. y x 1
10. y 3x 4
160
(3-40)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
11. x y 3
12. x y 1
21. y 3
22. y 1
13. 2x 3y 9
14. 3x 2y 6
Graph each linear inequality by using a test point. See Example 3.
23. 2x 3y 5
24. 5x 4y 3
15. 3x 4y 8
16. 4x 5y 10
25. x y 3 0
26. x y 6 0
17. x y 0
18. 2x y 0
1
1
27. x y 1
2
3
2
1
28. 2 y x
5
2
19. x 1
20. x 0
Graph each compound inequality. See Examples 4 and 5.
29. y x and y 2x 3
30. y x and y 3x 2
3.4
31. y x 3 or
y x 2
33. x y 5 and
xy3
35. x 2y 4 or
2x 3y 6
37. y 2 and x 3
32. y x 5 or
y 2x 1
Linear Inequalities and Their Graphs
(3-41)
161
39. y x and x 2
40. y x and y 0
41. 2x y 3 or
y2x
42. 3 x y 2 or
xy5
43. x 1 y x 3
44. x 1 y 2x 5
45. 0 y x and x 1
46. x y 1 and x 0
47. 1 x 3 and
2y5
48. 1 x 1 and
1 y 1
34. 2x y 3 and
3x y 0
36. 4x 3y 3 or
2x y 2
38. x 5 and y 1
162
(3-42)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
Graph the absolute value inequalities. See Example 6.
49. x y 2
50. 2x y 1
51. 2x y 1
53. x y 3 5
55. x 2y 4
57. x 2
59. y 1
60. y 2
61. y x 62. y x 3 63. x 2 and y 3
64. x 3 or y 1
65. x 3 1 and
y21
66. x 2 3 or
y5
2
52. x 2y 6
54. x 2y 4 2
56. x 3y 6
58. x 3
Solve each problem. See Example 7.
67. Budget planning. The Highway Patrol can spend a maximum of $120,000 on new vehicles this year. They can get a
fully equipped compact car for $15,000 or a fully equipped
full-size car for $20,000. Graph the region that shows the
number of cars of each type that could be purchased.
3.4
68. Allocating resources. A furniture maker has a shop that
can employ 12 workers for 40 hours per week at its maximum capacity. The shop makes tables and chairs. It takes
16 hours of labor to make a table and 8 hours of labor to
make a chair. Graph the region that shows the possibilities
for the number of tables and chairs that could be made in
one week.
69. More restrictions. In Exercise 67, add the condition that
the number of full-size cars must be greater than or equal to
the number of compact cars. Graph the region showing the
possibilities for the number of cars of each type that could
be purchased.
70. Chairs per table. In Exercise 68, add the condition that the
number of chairs must be at least four times the number of
tables and at most six times the number of tables. Graph the
region showing the possibilities for the number of tables
and chairs that could be made in one week.
71. Building fitness. To achieve cardiovascular fitness, you
should exercise so that your target heart rate is between
70% and 85% of its maximum rate. Your target heart
rate h depends on your age a. For building fitness, you
should have h 187 0.85a and h 154 0.70a
(NordicTrack brochure). Graph this compound inequality
Linear Inequalities and Their Graphs
(3-43)
163
for 20 a 75 to see the heart rate target zone for building fitness.
72. Waist-to-hip ratio. A study by Dr. Aaron R. Folsom concluded that waist-to-hip ratios are a better predictor of
5-year survival than more traditional height-to-weight ratios. Dr. Folsom concluded that for good health the waist
size of a woman aged 50 to 69 should be less than or
equal to 80% of her hip size, w 0.80h. Make a graph
showing possible waist and hip sizes for good health for
women in this age group for which hip size is no more
than 50 inches.
73. Advertising dollars. A restaurant manager can spend at
most $9000 on advertising per month and has two choices
for advertising. The manager can purchase an ad in the
Daily Chronicle (a 7-day-per-week newspaper) for $300
per day or a 30-second ad on WBTU television for $1000
each time the ad is aired. Graph the region that shows the
possible number of days that an ad can be run in the newspaper and the possible number of times that an ad can be
aired on television.
74. Shipping restrictions. The graph on the next page shows
all of the possibilities for the number of refrigerators and
the number of TVs that will fit into an 18-wheeler.
164
(3-44)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
a) Write an inequality to describe this region.
b) Will the truck hold 71 refrigerators and 118 TVs?
c) Will the truck hold 51 refrigerators and 176 TVs?
a) 3r t 330 b) no c) yes
400
Number of TVs
(0, 330)
300
200
GET TING MORE INVOLVED
100
75. Writing. Explain the difference between a compound inequality using the word and and a compound inequality
using the word or.
76. Discussion. Explain how to write an absolute value
inequality as a compound inequality.
(110, 0)
0
0
50
100
150
Number of refrigerators
FIGURE FOR EXERCISE 74
3.5
In this
section
●
Definitions
●
Domain and Range
●
The Rule Definition of
Function
●
Function Notation or
f-notation
●
Average Rate of Change
RELATIONS AND FUNCTIONS
Earlier in this chapter we used the phrase “is a function of ” to describe a special relationship between variables. The area of a circle is a function of its radius because
the area is determined by the radius using A r2. The area of rectangle is not a
function of its length because the area is not determined by the length alone. So “is
a function of” means “is determined by.” In this section we will learn that function
is also a noun. A function is a special kind of set. By studying functions as sets, we
can make the concept of functions more precise.
Definitions
Apple Imagewriter ribbons are sold in boxes of six in the K-LOG Catalog and are
priced as follows.
Number of boxes
Cost per ribbon
1
2–3
4
$4.85
$4.60
$4.35
We can write this data as a set of ordered pairs in which the first coordinate is the
number of boxes purchased and the second is the cost per ribbon in dollars:
(1, 4.85), (2, 4.60), (3, 4.60), (4, 4.35)
helpful
hint
The key word here is “determines.” According to the dictionary, determine means to
settle conclusively. If the second coordinate of an ordered
pair is inconclusive, then the
set of ordered pairs is not a
function.
Because the number of boxes determines the cost per ribbon, we say that the cost is
a function of the number of boxes purchased.
Suppose the following table appeared in the K-LOG Catalog:
Number of boxes
Cost per ribbon
1
2
2
4
$4.85
$4.60
$4.45
$4.35
Something is wrong with this table. The cost per ribbon when you buy two boxes is
not clear because the ordered pairs (2, 4.60) and (2, 4.45) have the same first coordinate and different second coordinates. In this case the cost per ribbon is not a
function of the number of boxes.
These examples illustrate the definition of function.
Function—A Set of Ordered Pairs
A function is a set of ordered pairs in which no two ordered pairs have the
same first coordinate and different second coordinates.