Document 1804649

Chapter 2
Lesson 2.1
Prerequisite Skills (p. 70)
1. A linear equation in one variable is an equation that can
be written in the form y 5 ax 1 b where a and b are
constants and a Þ 0.
2.1 Guided Practice (pp. 73–76)
1. a. The domain consists of all the x-coordinates: 24, 22,
0, and 1.
2. The absolute value of a real number is the distance the
The range consists of all the y-coordinates: 24, 22, 1,
and 3.
number is from zero on a number line.
3. 22(x 1 1) when x 5 25
b.
22(25 1 1) 5 22(24) 5 8
4. 11x 2 14 when x 5 23
11(23) 2 14 5 233 2 14 5 247
5. x 2 1 x 1 1 when x 5 4
42 1 4 1 1 5 16 1 4 1 1 5 21
6. 2x 2 2 3x 1 7 when x 5 1
2(12) 2 3(1) 1 7 5 21 2 3 1 7 5 3
7. 5x 2 2 5 8
8. 26x 2 10 5 20
5x 5 10
26x 5 30
x52
Check:
Check:
5x 2 2 5 8
5(2) 2 2 0 8
26x 2 10 5 20
26(25) 2 10 0 20
10 2 2 0 8
30 2 10 0 20
8 5 8 9. 2x 1 9 5 2x 2 27
9 5 3x 2 27
36 5 3x
12 5 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
x 5 25
20 5 20 Check:
2x 1 9 5 2x 2 27
212 1 9 0 2(12) 2 27
23 0 24 2 27
x
y
24
3
22
24
22
1
0
3
1
22
Input
Output
24
22
0
1
24
22
1
3
2. The relation is a function because each input is mapped
onto exactly one output.
3. Yes it is still a function because Kevin Garnett’s age is
the input and each age is mapped onto exactly one
average point.
4. y 5 3x 2 2
x
22
21
y
28
25
1
0
1
2
22 1
4
y
21
x
23 5 23 10. 2x 1 3y 5 6
3y 5 6 2 2x
6 2 2x
y5}
3
6
2x
y 5 }3 2 }
3
2
y 5 2 2 }3 x
12. x 1 4y 5 25
4y 5 25 2 x
25 2 x
y5}
4
5
x
y 5 2}4 2 }4
5
1
y 5 2}4 2 }4 x
11. 2x 2 y 5 10
2y 5 10 1 x
y 5 2(10 1 x)
y 5 210 2 x
5. The function f is not linear because it has an x 3-term.
f (x) 5 x 2 1 2 x 3
f (22) 5 (22) 2 1 2 (22)3 5 23 1 8 5 5
6. The function g is linear because it has the form
g(x) 5 mx 1 b.
g(x) 5 24 2 2x
g(22) 5 24 2 2(22) 5 24 1 4 5 0
7. Because the depth varies from 0 feet to 35,800 feet, a
reasonable domain is 0ada
The minimum value of P(d) 5 1, and the maximum
value of P(d) is P(35,800) 5 1075. So, a reasonable
range is 1ap (d)a1075.
2.1 Exercises (pp. 76–79)
Skill Practice
1. In the equation y 5 x 1 5, x is the independent variable
and y is the dependent variable.
2. The domain of a set of ordered pairs is all the
x-coordinates, and the range is all the y-coordinates.
Algebra 2
Worked-Out Solution Key
45
continued
3. (22, 3), (1, 2), (3, 21), (24, 23)
8. (4, 22), (4, 2), (16, 24), (16, 4)
The domain consists of all the x-coordinates: 24, 22, 1,
and 3.
The range consists of all the y-coordinates: 23, 21, 2,
and 3.
y
Input
Output
24
22
1
3
23
21
2
3
1
x
21
4. (5, 22), (23, 22), (3, 3), (21, 21)
The domain consists of all the x-coordinates: 23, 21, 3,
and 5.
The range consists of all the y-coordinates: 22, 21,
and 3.
y
1
x
22
Input
Output
23
21
3
5
22
21
3
The domain consists of all the x-coordinates: 22, 1,
and 6.
The range consists of all the y-coordinates: 23, 21, 5,
and 8.
Input
Output
22
23
21
5
8
1
2
6
x
22
6. (27, 4), (2, 25), (1, 22), (23, 6)
The domain consists of all the x-coordinates: 27, 23, 1,
and 2.
The range consists of all the y-coordinates: 25, 22, 4,
and 6.
y
2
22
x
Input
Output
27
23
1
2
25
22
4
6
7. (5, 20), (10, 20), (15, 30), (20, 30)
The range consists of all the y-coordinates: 20 and 30.
Input
10
25
46
Algebra 2
Worked-Out Solution Key
x
Input
y
4
1
2
x
16
Output
24
22
2
4
9. B; (24, 2), (21, 23), (1, 4), (1, 23), and (2, 1)
The domain consists of all the x-coordinates: 24, 21, 1,
and 2.
10. Yes; The relation is a function because each input is
mapped onto exactly one output.
11. Yes; The relation is a function because each input is
mapped onto exactly one output.
12. No; The relation is not a function because the input 21
is mapped onto both 2 and 21, and the input 5 is mapped
onto both 4 and 23.
mapped onto exactly one output.
14. The student incorrectly concludes that the relation is not
a function because more than one input is mapped onto
the same output.
The relation given by the ordered pairs (24, 2),
(21, 5), (3, 6), and (7, 2) is a function because each input
is mapped onto exactly one output.
15. The x-values are the inputs and the y-values are the
outputs. There should be one value of y for each value
of x.
The relation given by the table is not a function because
the input 0 is mapped onto both 5 and 9, and input 1 is
mapped onto both 6 and 8.
16. The relation given by the ordered pairs (3, 22), (0, 1),
(1, 0), (22, 21), (2, 21) is a function because each
imput is mapped onto exactly one output.
17. The relation given by the ordered pairs (2, 25), (22, 5),
(21, 4), (22, 0), and (3, 24) is not a function because
the input 22 is mapped onto both 0 and 5.
18. The relation given by the ordered pairs (0, 1), (1, 0),
(2, 3), (3, 2), and (4, 4) is a function because each input
is mapped onto exactly one output.
19. The relation given by the ordered pairs (21, 21), (2, 5),
The domain consists of all the x-coordinates: 5, 10, 15,
and 20.
y
The range consists of all the y-coordinates: 24, 22, 2,
and 4.
13. Yes; The relation is a function because each input is
5. (6, 21), (22, 23), (1, 8), (22, 5)
y
The domain consists of all the x-coordinates: 4 and 16.
5
10
15
20
utp
O
20
30
(4, 8), (25, 29), and (21, 25) is not a function because
the input 21 is mapped onto both 25 and 21.
20. B;
x
26
22
1 4 6
y
3
4
5 0 3
Of the ordered pairs to choose from, (6, 3) is the one
possible option to make a new function because the input
6 is the only input not in the previous ordered pairs.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Chapter 2,
Chapter 2,
21.
continued
22.
y
30. y 5 23x 1 2
y
2
x
1
2
x
1
The relation is a function.
23.
x
The relation is a function.
The relation is not a function.
y
1
y
22
y
0
21
1
1
2
x
21
8
5
2 21
24
31. y 5 22x
2
x
22
21
y
4
2
0
1
2
x
21
0 22 24
x
1
1
32. y 5 } x 1 2
2
x
24. If a vertical line intersects the graph more than once, it
means that for one x-value there is more than one y-value.
Because x is the input variable, there must be only one
y-value for each x-value for the relation to be a function.
25. y 5 x 1 2
x
y
22
y
21
0
1
0 1 2
2 3 4
y
x
x
21
y
22 21 0 1 2
3
2
1
1
5
2 }2 3
}
y
22 21
1
}
2
1
2}4
x
21
3
33. y 5 2} x 2 1
4
1
26. y 5 2x 1 5
y
0
1
1
2
22
x
7
5
21 2}4 2}2
y
x
22
21
0 1 2
y
7
6
5 4 3
34. The function f is linear because it has the form
f (x) 5 mx 1 b.
1
x
21
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
y
27. y 5 3x 1 1
22
21
0 1 2
y
25
22
1 4 7
f (8) 5 8 1 15 5 23
35. The function f is not linear because it has an x 2-term.
y
x
f (x) 5 x 1 15; f (8)
f (x) 5 x 2 1 1; f (23)
f (23) 5 (23)2 1 1 5 9 1 1 5 10
1
x
21
36. The function f is not linear because it has an {x{-term.
f (x) 5 {x{ 1 10; f (24)
28. y 5 5x 2 3
f (24) 5 {24{ 1 10 5 4 1 10 5 14
y
x
22
21
y
213
28 23 2 7
0
37. The function f is linear because it has the form
1 2
1
f (x) 5 mx 1 b, where m 5 0 and b 5 6.
x
21
f (x) 5 0x 1 6; f (2)
f (2) 5 0(2) 1 6 5 0 1 6 5 6
38. The function g is not linear because is has an x 3- and
x 2-term.
29. y 5 2x 2 7
x
y
22
21
0
1
2
211 29 27 25 23
1
21
y
g(x) 5 x 3 2 2x 2 1 5x 2 8; g(25)
x
g(25) 5 (25) 3 2 2(25) 2 1 5(25) 2 8
5 2125 2 50 2 25 2 8 5 2208
39. The function h is linear because it has the form
h(x) 5 mx 1 b.
2
h(x) 5 7 2 }3 x; h(15)
2
h(15) 5 7 2 }3 (15) 5 7 2 10 5 23
40. The equation y 5 {x{ is a function because each input
has exactly one output.
Algebra 2
Worked-Out Solution Key
47
continued
f (0) 5 f (0 1 0) 5 f (0) 1 f (0) 5 2f (0)
2f (0) 2 f (0) 5 0
h(t) 5 1000t 1 5400
4700 5 1000t
Problem Solving
4.7 5 t
42. The ordered pairs do not represent a function. The
x-values of 24, 25, and 26 each have two outputs.
43. The ordered pairs represent a function. For each x-value
there is exactly one output.
44. V(s) 5 s 3;
V(4) 5 43 5 64 units3
V(4) represents the volume of a cube when the length of
its side is 4 units.
4
45. V(r) 5 } : r 3
3
4
V(6) 5 }3 : (63) 5 }3 :(216) 5 288: ø 904.8 units 3
V(6) represents the volume of a sphere with a radius of 6
units.
46. Domain: 0ata5, Range: 140.7 a w(t) a 172;
Over the years 1999–2004 the watermelon acreage
ranged from a low of 140,700 in 2004 to a high of
172,000 in 1999.
Height (inches)
The minimum value of h(*) is h(15) 5 57.95, and
the maximum value of h(*) is h(24) 5 75.5. So, a
reasonable range is 57.95ah(*)a75.5.
h(l)
74
72
70
68
66
64
62
60
58
0
9000
The graph of h(t) is
shown. Because the
time varies from 0
hours to 4.7 hours, a
reasonable domain
is 0ata4.7.
8000
7000
6000
5000
0
0
1
2
3
Time (hours)
4
t
The minimum value of h(t) is h(0) 5 5400, and the
maximum value of h(t) is h(4.7) 5 10,100. So a
reasonable range is 5400ah(t)a10,100.
At the time of 3.5 hours the elevation of the climber is
h(3.5) 5 1000(3.5) 1 5400 5 8900 feet, which you can
verify from the graph.
49. a. domain: 11,350,000; 12,280,000; 12,420,000;
15,980,000; 18,980,000; 20,850,000; 33,870,000
range: 20, 21, 27, 31, 34, 55
b. Yes, each input has exactly one output.
c. No, input 21 has more than one output.
50. a. Yes, it is a function, because each merchandise cost is
mapped onto exactly one shipping cost.
47. a. The graph of h(*) is shown. Because the length varies
from 15 inches to 24 inches, a reasonable domain
is 15a*a24.
h(t)
10,000
Camp Muir is:
10,100 5 1000t 1 5400
f (0) 5 0
4
48. The time to get to
b. No, it is not a function, because each shipping cost
value is mapped onto a range of merchandise cost
values.
Mixed Review for TAKS
51. D;
b 5 2t
b 5 210 5 1024
After 10 hours, there are 1024 billion bacteria.
52. H;
1
1
Area 5 15(8) 1 }2(6)(7) 1 }2(6)(7) 5 162 cm2
0 15
17
19
21
Length (inches)
23 l
b. At a length of 15.5 inches the height of the adult
female was h(15.5) 5 1.95(15.5) 1 28.7 ø 58.9
inches, which you can verify by the graph.
c. 5 feet 11 inches 5 71 inches
At a height of 71 inches the length of the femur is
The area of the figure is 162 square centimeters.
2.1 Extension (p. 81)
1. y 5 2x 1 3; domain: 22, 21, 0, 1, 2
x
22
y
21
y
21 0 1 2
1
3 5 7
71 5 1.95* 1 28.7
42.3 5 1.95*
21.7 ø *
The femur is about 21.7 inches long.
1
21
x
The graph consists of separate points, so the function is
discrete. Its range is 21, 1, 3, 5, and 7.
48
Algebra 2
Worked-Out Solution Key
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
41. f (2a) 5 f (a 1 a) 5 f (a) 1 f (a) 5 2 + f (a)
Elevation (feet)
Chapter 2,
Chapter 2,
continued
2. f (x) 5 0.5x 2 4; domain: 24, 22, 0, 2, 4
24 22
y
26 25 24 23 22
0
2
4
1
m(x) 5 3x. The first
eleven points of the
graph m(x) are
shown. Because milk
is only delivered
once a week, only
whole numbers can
be used for each week.
The domain is
0 1 2 3 4 5 6 7 8 9 10
Weeks
the set of whole
numbers 0, 1, 2, 3, . . . From the graph, you can see that
the range is 0, 3, 6, 9, . . . The graph consists of separate
points, so the function is discrete.
y
x
21
The graph consists of separate points, so the function is
discrete. Its range is 26, 25, 24, 23, and 22.
3. y 5 23x 1 9; domain: x < 5
21 0 1 2
y
12
5
1
y
8. The function is
The graph is unbroken, so the function is continuous.
Its range is y > 26.
1
4. f (x) 5 } x 1 6; domain: xq26
3
x
26
y
4
6 7 8
x
The graph is unbroken, so the function is continuous. Its
range is y q4.
Distance (miles)
d(x) 5 3.5x. Amanda
can walk any nonnegative
amount of time, so the
domain is xq0. From
the graph you can see that
the range is yq0. The
graph is unbroken, so
the function is continuous.
x
2.2 Guided Practice (pp. 83–85)
14
1. The skateboard ramp has a rise of 12 inches and a run of
54 inches.
10
rise
12
2
Slope 5 }
5 }9
run 5 }
54
6
2
2
0
6. The function is
1
21
Lesson 2.2
d(x)
The slope of the ramp is }9.
0
1
2
3
Hours
4
x
s(x) 5 1.25x. The first
nine points of the graph
8
s(x) are shown. The
6
number of times the
subway is ridden must
4
be a whole number, so
the domain is the set of
2
whole numbers
0
0, 1, 2, 3, . . . From the
0
2
4
6
8
Number of rides
graph, you can see that
the range is 0, 1.25, 2.50, 3.75, 5.00, . . . The graph
consists of separate points, so the function is discrete.
2. D;
Let (x1, y1)5 (24, 9) and (x2, y2) 5 (28, 3).
s(x)
10
y2 2 y1
329
26
3
m5}
5}
=}
x 2x 5}
24 2
28 2 (24)
2
1
3. Let (x1, y1)5 (0, 3) and (x2, y2) 5 (4, 8).
y2 2 y1
823
5
m5}
5}
5 }4
x2 2 x1
420
Cost (dollars)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1
21
x
y
x can be any real number on
a number line, so the domain
is the set of all real numbers.
From the graph, you can see
that the range is the set of all
real numbers. The graph is
unbroken, so the function
is continuous.
y
5. The function is
10
20
30
40
Length of cable (feet)
9. The function is d(x) 5 x 2 3.
23 0 3 6
5
f(x)
w (x) 5 0.24x. The steel
12
cable can be any
8
nonnegative amount
of length in feet, so the
4
domain is xq0. From
the graph, you can see
0
0
that the range is yq0.
The graph is unbroken,
so the function is continuous.
x
21
9 6 3 26
x
Weight (pounds)
x
m(x)
30
27
24
21
18
15
12
9
6
3
0
Gallons of milk
x
7. The function is
4. Let (x1, y1) 5 (25, 1) and (x2, y2) 5 (5, 24).
y2 2 y1
25
24 2 1
1
m5}
5}
5}
5 2}2
x2 2 x1
10
5 2 (25)
x
5. Let (x1, y1) 5 (23, 22) and (x2, y2) 5 (6, 1).
y2 2 y1
1 2 (22)
3
1
m5}
5}
5 }9 5 }3
x2 2 x1
6 2 (23)
Algebra 2
Worked-Out Solution Key
49