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Graph each function. Identify the domain and
range.
1.
SOLUTION:
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
The y-coordinates of points on the graph are real
numbers less than or equal to 4, so the range
is .
2-6 Special Functions
Graph each function. Identify the domain and
range.
2.
1.
SOLUTION:
SOLUTION:
The function is defined for all real values of x, so the
domain is all real numbers.
The function is defined for all real values of x, so the D = {all real numbers}
domain is all real numbers.
The y-coordinates of points on the graph are real
D = {all real numbers} numbers between 8 and –2 and less than or equal to
–8, so the range is .
The y-coordinates of points on the graph are real
numbers less than or equal to 4, so the range Write the piecewise-defined function shown in
is .
each graph.
2.
3.
SOLUTION:
SOLUTION:
The left portion of the graph is the line g(x) = x + 4.
There is an open circle at (–2, 2), so the domain for
this part of the function is .
The center portion of the graph is the constant
function g(x) = –3. There are closed dots at (–2, –3)
The function is defined for all real values of x, so the and (3, 3), so the domain for this part is
domain is all real numbers.
.
D = {all real numbers} The right portion of the graph is the line g(x) = –2x +
12. There is an open circle at (3, 6), so the domain
The y-coordinates of points on the graph are real for this part is .
eSolutions Manual - Powered by Cognero Page1
numbers between 8 and –2 and less than or equal to
–8, so the range is . Write the piecewise function.
Write the piecewise-defined function shown in
each graph.
3.
SOLUTION:
The left portion of the graph is the line g(x) = x + 4. 4.
There is an open circle at (–2, 2), so the domain for
this part of the function is .
SOLUTION:
The center portion of the graph is the constant The left portion of the graph is the constant function
function g(x) = –3. There are closed dots at (–2, –3) g(x) = 6. There is a closed dot at (–5, 6), so the
and (3, 3), so the domain for this part is domain for this part is .
. The center portion of the graph is the line g(x) = –x
The right portion of the graph is the line g(x) = –2x + + 4. There are open circles at (–5, 9) and (–2, 6), so
12. There is an open circle at (3, 6), so the domain the domain for this part is .
for this part is .
The right portion of the graph is the line
Write the piecewise function. . There is a closed dot at (–2, 0), so
the domain for this part is .
Write the piecewise function.
CCSS REASONING
5. Springfield High School’s
theater can hold 250 students. The drama club is
performing a play in the theater. Draw a graph of a
4. step function that shows the relationship between the
number of tickets sold x and the minimum number of
performances y that the drama club must do.
SOLUTION:
The left portion of the graph is the constant function
g(x) = 6. There is a closed dot at (–5, 6), so the
SOLUTION:
domain for this part is . When x is greater than 0 and less than or equal to
250, the drama club needs to do only one
The center portion of the graph is the line g(x) = –x performance. When x is greater than 250 and less
+ 4. There are open circles at (–5, 9) and (–2, 6), so than or equal to 500, they must do at least two
the domain for this part is . performances. Continue the pattern with a table.
The right portion of the graph is the line
. There is a closed dot at (–2, 0), so
the domain for this part is .
Write the piecewise function.
CCSS REASONING
5. Springfield High School’s
theater can hold 250 students. The drama club is
performing a play in the theater. Draw a graph of a
step function that shows the relationship between the
number of tickets sold x and the minimum number of Graph each function. Identify the domain and
performances y that the drama club must do.
range.
SOLUTION: 6.
When x is greater than 0 and less than or equal to
250, the drama club needs to do only one
performance. When x is greater than 250 and less SOLUTION:
than or equal to 500, they must do at least two
performances. Continue the pattern with a table.
D = {all real numbers}
The function g(x) is a reflection of twice of a
greatest integer function. So, g(x) takes all even
integer values or zero.
R = {all even integers}
7.
Graph each function. Identify the domain and SOLUTION:
range.
6.
SOLUTION:
D = {all real numbers}
R = {all integers}
Graph each function. Identify the domain and
D = {all real numbers} range.
The function g(x) is a reflection of twice of a 8.
greatest integer function. So, g(x) takes all even
integer values or zero.
SOLUTION:
R = {all even integers}
7.
SOLUTION:
D = {all real numbers}
9.
D = {all real numbers}
SOLUTION:
R = {all integers}
Graph each function. Identify the domain and
range.
8.
SOLUTION: D = {all real numbers}
.
10.
D = {all real numbers} SOLUTION:
9.
SOLUTION: D = {all real numbers}
.
11.
D = {all real numbers}
SOLUTION:
.
10.
SOLUTION: D = {all real numbers}
.
Graph each function. Identify the domain and
range.
D = {all real numbers}
12.
.
SOLUTION:
11.
SOLUTION:
.
.
D = {all real numbers}
.
13.
Graph each function. Identify the domain and
range.
SOLUTION:
12.
SOLUTION:
.
.
. 14.
.
SOLUTION:
13.
SOLUTION:
D = {all real numbers}
.
15.
.
. SOLUTION:
14.
SOLUTION:
.
Write the piecewise-defined function shown in
each graph.
D = {all real numbers}
.
16.
SOLUTION:
15. The left portion of the graph is the constant function
g(x) = –8. There is a closed dot at (–6, –8), so the
domain for this part of the function is .
SOLUTION:
The center portion of the graph is the line g(x) =
0.25x + 2. There are closed dots at (–4, 1) and (4, 3),
so the domain for this part is .
The right portion of the graph is the constant function
g(x) = 4. There is an open circle at (6, 4), so the
constant function is defined for .
. Write the piecewise function.
Write the piecewise-defined function shown in
each graph.
16.
SOLUTION:
17.
The left portion of the graph is the constant function
g(x) = –8. There is a closed dot at (–6, –8), so the
domain for this part of the function is . SOLUTION:
The left portion of the graph is the line g(x) = –x – 4.
The center portion of the graph is the line g(x) = There is an open circle at (–3, –1), so the domain for
0.25x + 2. There are closed dots at (–4, 1) and (4, 3), this part of the function is
so the domain for this part is . .
The center portion of the graph is the line g(x) = x +
The right portion of the graph is the constant function 1. There are closed dots at (–3, –2) and (1, 2), so the
g(x) = 4. There is an open circle at (6, 4), so the domain for this part is .
constant function is defined for .
The right portion of the graph is the constant function
Write the piecewise function. g(x) = –6. There is an open circle at (4, –6), so the
domain for this part is .
Write the piecewise function.
17.
18.
SOLUTION:
The left portion of the graph is the line g(x) = –x – 4. SOLUTION:
There is an open circle at (–3, –1), so the domain for The left portion of the graph is the constant function
this part of the function is g(x) = –9. There is an open circle at (–5, –9), so the
. domain for this part of the function is .
The center portion of the graph is the line g(x) = x +
1. There are closed dots at (–3, –2) and (1, 2), so the
domain for this part is . The center portion of the graph is the line g(x) = x +
4. There are closed dots at (0, 4) and (3, 7), so the
The right portion of the graph is the constant function domain for this part is .
g(x) = –6. There is an open circle at (4, –6), so the The right portion of the graph is the line g(x) = x – 3.
domain for this part is . There is an open circle at (7, 4), so the domain for
Write the piecewise function. this part is .
Write the piecewise function.
18.
19.
SOLUTION:
The left portion of the graph is the constant function
g(x) = –9. There is an open circle at (–5, –9), so the SOLUTION:
domain for this part of the function is . The left portion of the graph is the constant function
g(x) = 8. There is a closed dot at (–1, 8), so the
domain for this part is .
The center portion of the graph is the line g(x) = x +
4. There are closed dots at (0, 4) and (3, 7), so the The center portion of the graph is the line g(x) = 2x.
domain for this part is . There are closed dots at (4, 8) and (6, 12), so the
The right portion of the graph is the line g(x) = x – 3. domain for this part is .
There is an open circle at (7, 4), so the domain for
this part is . The right portion of the graph is the line g(x) = 2x –
15. There is a circle at (7, –1), so the domain for this
Write the piecewise function. part is .
Write the piecewise function.
Graph each function. Identify the domain and
range.
20.
19.
SOLUTION:
SOLUTION:
The left portion of the graph is the constant function
g(x) = 8. There is a closed dot at (–1, 8), so the
domain for this part is .
The center portion of the graph is the line g(x) = 2x.
There are closed dots at (4, 8) and (6, 12), so the
domain for this part is .
D = {all real numbers}
The right portion of the graph is the line g(x) = 2x –
15. There is a circle at (7, –1), so the domain for this R = {all integers}
part is .
Write the piecewise function.
21.
SOLUTION:
Graph each function. Identify the domain and
range.
20.
D = {all real numbers}
SOLUTION:
R = {all integers}
22.
SOLUTION:
D = {all real numbers}
R = {all integers}
21.
D = {all real numbers}
SOLUTION:
R = {all integers}
23.
SOLUTION:
D = {all real numbers}
R = {all integers}
22.
The function is defined for all real values of x, so the
domain is all real numbers.
SOLUTION:
D = {all real numbers}
The function g(x) is twice of a greatest integer
function. So, g(x) takes only even integer values.
Therefore, the range is R = {all even integers}.
Graph each function. Identify the domain and
D = {all real numbers} range.
R = {all integers} 24.
SOLUTION:
23.
SOLUTION:
D = {all real numbers}
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers} 25.
The function g(x) is twice of a greatest integer
function. So, g(x) takes only even integer values. SOLUTION:
Therefore, the range is R = {all even integers}.
Graph each function. Identify the domain and
range.
24.
SOLUTION: D = {all real numbers
}
26.
SOLUTION:
D = {all real numbers}
25.
D = {all real numbers}
SOLUTION:
27.
SOLUTION:
D = {all real numbers
}
26.
D = {all real numbers}
SOLUTION:
28.
SOLUTION:
D = {all real numbers}
27.
D = {all real numbers}
SOLUTION:
29.
SOLUTION:
D = {all real numbers}
28.
D = {all real numbers}
SOLUTION:
30. GIVING Patrick is donating money and volunteering
his time to an organization that restores homes for
the needy. His employer will match his monetary
donations up to $100
D = {all real numbers} a. Identify the type of function that models the total
money received by the charity when Patrick donates
x dollars.
b. Write and graph a function for the situation.
29.
SOLUTION:
a
. The function is composed of two distinct linear
SOLUTION: functions. Therefore, it is a piecewise function.
b.
D = {all real numbers}
30. GIVING Patrick is donating money and volunteering
his time to an organization that restores homes for
the needy. His employer will match his monetary
donations up to $100 31. CCSS SENSE-MAKINGA car’s speedometer
reads 60 miles an hour.
a. Identify the type of function that models the total
money received by the charity when Patrick donates a. Write an absolute value function for the difference
x dollars. between the car’s actual speed a and the reading on
the speedometer.
b. Write and graph a function for the situation.
b. What is an appropriate domain for the function?
Explain your reasoning.
SOLUTION:
a
. The function is composed of two distinct linear
functions. Therefore, it is a piecewise function. c. Use the domain to graph the function.
SOLUTION:
b. a. The absolute value function is .
b. Since the speed of the car cannot be negative, the
appropriate domain for the function is {a | a ≥ 0}.
c.
31. CCSS SENSE-MAKINGA car’s speedometer
reads 60 miles an hour.
a. Write an absolute value function for the difference
between the car’s actual speed a and the reading on 32. RECREATIONThe charge for renting a bicycle
the speedometer. from a rental shop for different amounts of time is
shown at the right.
b. What is an appropriate domain for the function? a. Identify the type of function that models this
Explain your reasoning. situation.
c. Use the domain to graph the function. b. Write and graph a function for the situation.
SOLUTION:
a. The absolute value function is .
b. Since the speed of the car cannot be negative, the
appropriate domain for the function is {a | a ≥ 0}.
c.
SOLUTION:
a. The rent is constant in each interval. Therefore,
the situation is best modeled by a step function.
b.
32. RECREATIONThe charge for renting a bicycle
from a rental shop for different amounts of time is
shown at the right.
a. Identify the type of function that models this
situation.
b. Write and graph a function for the situation.
Use each graph to write the absolute value
function.
SOLUTION:
a. The rent is constant in each interval. Therefore,
the situation is best modeled by a step function.
b.
33.
SOLUTION:
The graph changes its direction at (0, 0).
The slope of the line in the interval is –0.5.
The slope of the line in the interval is 0.5
.
Therefore, the absolute value function is
.
Use each graph to write the absolute value
function.
34.
SOLUTION:
The graph changes its direction at (–5, –4).
The slope of the line in the interval 1.
is –
33.
The slope of the line in the interval
is 1.
SOLUTION:
The graph changes its direction at (0, 0). Therefore, the absolute value function is
.
The slope of the line in the interval is –0.5.
Graph each function. Identify the domain and
The slope of the line in the interval is 0.5
. range.
Therefore, the absolute value function is
. 35.
SOLUTION:
34.
SOLUTION: D = {all real numbers}
The graph changes its direction at (–5, –4).
The slope of the line in the interval 1.
is –
The slope of the line in the interval
is 1.
36.
Therefore, the absolute value function is
.
SOLUTION:
Graph each function. Identify the domain and
range.
35.
SOLUTION: D = {all real numbers};
R = {all non-negative integers}
37.
D = {all real numbers}
SOLUTION:
36.
SOLUTION:
D = {all real numbers}
.
38.
D = {all real numbers};
R = {all non-negative integers}
SOLUTION:
37.
SOLUTION:
D = {all real numbers}
MULTIPLEREPRESENTATIONS
39. Consider the
following absolute value functions.
D = {all real numbers}
.
a. TABULAR
Use a graphing calculator to create a
table of f (x) and g(x) values for x = 4 to x = 4.
–
b. GRAPHICAL
Graph the functions on separate
38. graphs.
c. NUMERICAL
Determine the slope between
each two consecutive points in the table.
SOLUTION:
d. VERBAL
Describe how the slopes of the two
sections of an absolute value graph are related.
SOLUTION:
a.
D = {all real numbers}
b.
MULTIPLEREPRESENTATIONS
39. Consider the
following absolute value functions.
a. TABULAR c.
Use a graphing calculator to create a
table of f (x) and g(x) values for x = 4 to x = 4.
–
b. GRAPHICAL
Graph the functions on separate
graphs.
c. NUMERICAL
Determine the slope between
each two consecutive points in the table. d. The two sections of an absolute value graph have
opposite slopes. The slope is constant for each
d. VERBAL section of the graph.
Describe how the slopes of the two
sections of an absolute value graph are related.
OPENENDED
40. Write an absolute value relation in
SOLUTION: which the domain is all nonnegative numbers and the
a. range is all real numbers.
SOLUTION:
Sample answer:
| y | = x
b.
CHALLENGEGraph
41.
SOLUTION:
c.
CCSS ARGUMENTSFind a counterexample to
42.
the following statement and explain your reasoning.
d. The two sections of an absolute value graph have In order to find the greatest integer function of x
opposite slopes. The slope is constant for each when x is not an integer, round x to the nearest
section of the graph.
integer.
SOLUTION:
OPENENDED
40. Write an absolute value relation in
which the domain is all nonnegative numbers and the Sample answer: 8.6
range is all real numbers. The greatest integer function asks for the greatest
integer less than or equal to the given value; thus 8 is
the greatest integer. If we were to round this value to
SOLUTION: the nearest integer, we would round up to 9.
Sample answer:
| y | = x
OPENENDED
43. Write an absolute value function in
CHALLENGEGraph
41. which f(5) = 3.
–
SOLUTION:
SOLUTION:
Sample answer:
WRITING INMATH
44. Explain how piecewise
functions can be used to accurately represent real-
world problems.
SOLUTION:
Sample answer:
Piecewise functions can be used to
CCSS ARGUMENTSFind a counterexample to represent the cost of items when purchased in
42.
the following statement and explain your reasoning. quantities, such as a dozen eggs.
In order to find the greatest integer function of x
when x is not an integer, round x to the nearest
integer. SHORT RESPONSEWhat expression gives the
45.
nth term of the linear pattern defined by the table?
SOLUTION:
Sample answer: 8.6
The greatest integer function asks for the greatest
integer less than or equal to the given value; thus 8 is
the greatest integer. If we were to round this value to
the nearest integer, we would round up to 9. SOLUTION:
OPENENDED
43. Write an absolute value function in So, the nth term is 3n + 1.
which f(5) = 3.
–
SOLUTION: Solve: 5(x + 4) = x + 4
46.
Sample answer:
Step 1: 5x + 20 = x + 4
WRITING INMATH Step 2: 4x + 20 = 4
44. Explain how piecewise
functions can be used to accurately represent real-
x = 24
world problems. Step 3: 4
x = 6
Step 4:
SOLUTION:
Sample answer: Piecewise functions can be used to Which is the first incorrect step in the solution
represent the cost of items when purchased in shown above?
quantities, such as a dozen eggs.
A Step 4
SHORT RESPONSEWhat expression gives the
45. B Step 3
nth term of the linear pattern defined by the table?
C Step 2
D Step 1
SOLUTION:
SOLUTION:
So, the nth term is 3n + 1.
46. Solve: 5(x + 4) = x + 4
Compare the steps. The first incorrect step in the
Step 1: 5x + 20 = x + 4 solution is on step 3.
Therefore, option B is the correct answer.
Step 2: 4x + 20 = 4
x = 24
Step 3: 4
NUMBERTHEORY Twelve consecutive integers
47.
are arranged in order from least to greatest. If the
x = 6
Step 4: sum of the first six integers is 381, what is the sum of
the last six integers?
Which is the first incorrect step in the solution
shown above? F
345
AStep 4
G381
BStep 3 H 387
CStep 2 J
417
D Step 1
SOLUTION:
Let x be least number in the consecutive integer.
SOLUTION:
Sum of the first six integers = x + (x + 1) + (x + 2) +
(x + 3) + (x + 4) + (x + 5)
= 6x + 15
Equate 6x + 15 to 381 and solve for x.
Compare the steps. The first incorrect step in the
solution is on step 3.
Therefore, option B is the correct answer.
Therefore, the last 6 integers are 67, 68, 69, 70, 71
and 72.
NUMBERTHEORY Twelve consecutive integers
47. 67 + 68 + 69 + 70 + 71 + 72 = 417
are arranged in order from least to greatest. If the
sum of the first six integers is 381, what is the sum of
the last six integers? Therefore, option J is the correct answer.
F
345 ACT/SAT For which function does
48.
G381
H 387
A
J
417
B
SOLUTION:
Let x be least number in the consecutive integer.
C
Sum of the first six integers = x + (x + 1) + (x + 2) +
(x + 3) + (x + 4) + (x + 5) D
= 6x + 15
Equate 6x + 15 to 381 and solve for x.
E
SOLUTION:
Therefore, the last 6 integers are 67, 68, 69, 70, 71
and 72.
67 + 68 + 69 + 70 + 71 + 72 = 417
Therefore, option J is the correct answer.
ACT/SAT For which function does
48.
Therefore, option B is the correct answer.
FOOTBALLThe table shows the relationship
A 49.
between the total number of male students per school
and the number of students who tried out for the
B football team.
C a. Find a regression equation for the data.
b. Determine the correlation coefficient.
D
c. Predict how many students will try out for football
E at a school with 800 male students.
SOLUTION:
SOLUTION:
a.
y = 0.10x + 30.34
b. r = 0.987
Therefore, option B is the correct answer.
c. Substitute x = 800 in the equation y = 0.10x +
30.34.
FOOTBALLThe table shows the relationship
49.
between the total number of male students per school
and the number of students who tried out for the
football team.
a. Find a regression equation for the data.
b. Determine the correlation coefficient.
So, at a school with 800 male students, about 110
students will try out for football.
c. Predict how many students will try out for football
at a school with 800 male students.
Write an equation in slope-intercept form for
the line described.
passes through ( 3, 6), perpendicular to y = 2x + 1
50. – – –
SOLUTION:
The slope of the line y = 2x + 1 is 2.
– –
Therefore, the slope of a line perpendicular to y =
–
2x + 1 is .
SOLUTION:
a.
y = 0.10x + 30.34 Substitute 0.5 for m in the slope-intercept form.
b. r = 0.987
c. Substitute x = 800 in the equation y = 0.10x + Substitute 3 and 6 for x and y and solve for b
30.34. – –
.
Therefore, the equation of the line which passes
through the point ( 3, 6) and is perpendicular to y =
So, at a school with 800 male students, about 110 – –
students will try out for football. –2x +1 is y = 0.5x – 4.5.
Write an equation in slope-intercept form for 51. passes through (4, 0), parallel to 3x + 2y = 6
the line described.
passes through ( 3, 6), perpendicular to y = 2x + 1 SOLUTION:
50. – – –
The slope of the line 3x + 2y = 6 is .
SOLUTION:
The slope of the line y = 2x + 1 is 2.
– – Therefore, the slope of a line parallel to the line 3x +
2y = 6 is .
Therefore, the slope of a line perpendicular to y =
–
2x + 1 is .
Substitute m in the slope-intercept form.
for
Substitute 0.5 for m in the slope-intercept form.
Substitute 3 and 6 for x and y and solve for b
– – Substitute 4 and 0 for x and y and solve for b.
.
Therefore, the equation of the line which passes
through the point (–3, –6) and is perpendicular to y =
2x +1 is y = 0.5x 4.5.
– – Therefore, the equation of the line which passes
through the point (4, 0) and is parallel to 3x + 2y
= 6
is .
51. passes through (4, 0), parallel to 3x + 2y = 6
SOLUTION: passes through the origin, perpendicular to 4x 3y =
52. –
The slope of the line 3x + 2y = 6 is . 12
Therefore, the slope of a line parallel to the line 3x + SOLUTION:
The slope of the line 4x 3y = 12 is .
2y = 6 is . –
Therefore, the slope of a line perpendicular to the line
Substitute m in the slope-intercept form.
for 4x 3y .
– = 12 is
Substitute m in the slope-intercept form.
for
Substitute 4 and 0 for x and y and solve for b.
Substitute 0 and 0 for x and y and solve for b.
Therefore, the equation of the line which passes
through the point (4, 0) and is parallel to 3x + 2y
= 6 Therefore, the equation of the line which passes
is . through the origin and is perpendicular to 4x 3y =
–
12 is .
52. passes through the origin, perpendicular to 4x – 3y =
12
2
Find each value if f (x) = 4x + 6, g(x) = x , and
– –
2
h(x) = 2x 6x + 9.
– –
SOLUTION:
The slope of the line 4x – 3y = 12 is . f (2c)
53.
Therefore, the slope of a line perpendicular to the line
SOLUTION:
4x 3y . Substitute 2c for x in the function f (x).
– = 12 is
Substitute m in the slope-intercept form.
for
g(a + 1)
54.
Substitute 0 and 0 for x and y and solve for b.
SOLUTION:
Substitute a + 1 for x in the function g(x).
Therefore, the equation of the line which passes
through the origin and is perpendicular to 4x – 3y =
12 is .
55. h(6)
2
Find each value if f (x) = 4x + 6, g(x) = x , and SOLUTION:
– –
2 Substitute 6 for x in the function h(x).
h(x) = 2x 6x + 9.
– –
f (2c)
53.
SOLUTION:
Substitute 2c for x in the function f (x).
56. Determine whether the figures below are similar.
g(a + 1)
54.
SOLUTION:
The ratio between the length of the rectangles is
SOLUTION:
Substitute a + 1 for x in the function g(x). .
The ratio between the width of the rectangles is
.
Since the ratios of the sides are equal, the given
h(6) figures are similar.
55.
SOLUTION: Graph each equation.
Substitute 6 for x in the function h(x).
y = 0.25x + 8
57. –
SOLUTION:
56. Determine whether the figures below are similar.
SOLUTION:
The ratio between the length of the rectangles is
. 58.
The ratio between the width of the rectangles is SOLUTION:
.
Since the ratios of the sides are equal, the given
figures are similar.
Graph each equation.
y = 0.25x + 8 59. 8x + 4y = 32
57. –
SOLUTION:
SOLUTION:
58.
SOLUTION:
59. 8x + 4y = 32
SOLUTION:
Graph each function. Identify the domain and
range.
1.
SOLUTION:
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
The y-coordinates of points on the graph are real
numbers less than or equal to 4, so the range
is .
2.
SOLUTION:
Graph each function. Identify the domain and
range.
1.
SOLUTION: The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
The y-coordinates of points on the graph are real
numbers between 8 and –2 and less than or equal to
–8, so the range is .
Write the piecewise-defined function shown in
The function is defined for all real values of x, so the
domain is all real numbers. each graph.
D = {all real numbers}
The y-coordinates of points on the graph are real
numbers less than or equal to 4, so the range
is .
3.
2. SOLUTION:
The left portion of the graph is the line g(x) = x + 4.
There is an open circle at (–2, 2), so the domain for
this part of the function is .
SOLUTION:
The center portion of the graph is the constant
function g(x) = –3. There are closed dots at (–2, –3)
and (3, 3), so the domain for this part is
.
The right portion of the graph is the line g(x) = –2x +
12. There is an open circle at (3, 6), so the domain
for this part is .
The function is defined for all real values of x, so the
domain is all real numbers. Write the piecewise function.
D = {all real numbers}
The y-coordinates of points on the graph are real
numbers between 8 and –2 and less than or equal to
2-6 Special Functions
–8, so the range is .
Write the piecewise-defined function shown in
each graph.
4.
SOLUTION:
3. The left portion of the graph is the constant function
g(x) = 6. There is a closed dot at (–5, 6), so the
SOLUTION: domain for this part is .
The left portion of the graph is the line g(x) = x + 4.
There is an open circle at (–2, 2), so the domain for The center portion of the graph is the line g(x) = –x
this part of the function is . + 4. There are open circles at (–5, 9) and (–2, 6), so
the domain for this part is .
The center portion of the graph is the constant
function g(x) = –3. There are closed dots at (–2, –3) The right portion of the graph is the line
and (3, 3), so the domain for this part is
. There is a closed dot at (–2, 0), so
. the domain for this part is .
The right portion of the graph is the line g(x) = –2x + Write the piecewise function.
12. There is an open circle at (3, 6), so the domain
for this part is .
Write the piecewise function.
CCSS REASONING
5. Springfield High School’s
theater can hold 250 students. The drama club is
performing a play in the theater. Draw a graph of a
step function that shows the relationship between the
number of tickets sold x and the minimum number of
performances y that the drama club must do.
SOLUTION:
When x is greater than 0 and less than or equal to
4. 250, the drama club needs to do only one
performance. When x is greater than 250 and less
than or equal to 500, they must do at least two
SOLUTION: performances. Continue the pattern with a table.
The left portion of the graph is the constant function
g(x) = 6. There is a closed dot at (–5, 6), so the
domain for this part is .
eSolutions Manual - Powered by Cognero Page2
The center portion of the graph is the line g(x) = –x
+ 4. There are open circles at (–5, 9) and (–2, 6), so
the domain for this part is .
The right portion of the graph is the line
. There is a closed dot at (–2, 0), so
the domain for this part is .
Write the piecewise function.
Graph each function. Identify the domain and
range.
CCSS REASONING
5. Springfield High School’s
theater can hold 250 students. The drama club is 6.
performing a play in the theater. Draw a graph of a
step function that shows the relationship between the
number of tickets sold x and the minimum number of SOLUTION:
performances y that the drama club must do.
SOLUTION:
When x is greater than 0 and less than or equal to
250, the drama club needs to do only one
performance. When x is greater than 250 and less
than or equal to 500, they must do at least two
performances. Continue the pattern with a table.
D = {all real numbers}
The function g(x) is a reflection of twice of a
greatest integer function. So, g(x) takes all even
integer values or zero.
R = {all even integers}
7.
SOLUTION:
Graph each function. Identify the domain and
range.
D = {all real numbers}
6.
R = {all integers}
SOLUTION:
Graph each function. Identify the domain and
range.
8.
SOLUTION:
D = {all real numbers}
The function g(x) is a reflection of twice of a
greatest integer function. So, g(x) takes all even
integer values or zero.
R = {all even integers}
D = {all real numbers}
7.
SOLUTION:
9.
SOLUTION:
D = {all real numbers}
R = {all integers}
Graph each function. Identify the domain and D = {all real numbers}
range.
.
8.
10.
SOLUTION:
SOLUTION:
D = {all real numbers}
D = {all real numbers}
.
9.
SOLUTION:
11.
SOLUTION:
D = {all real numbers}
.
D = {all real numbers}
10. .
Graph each function. Identify the domain and
SOLUTION:
range.
12.
SOLUTION:
D = {all real numbers}
.
11.
.
SOLUTION:
.
13.
D = {all real numbers}
SOLUTION:
.
Graph each function. Identify the domain and
range.
12.
.
.
SOLUTION:
14.
SOLUTION:
.
.
D = {all real numbers}
13.
.
SOLUTION:
15.
SOLUTION:
.
.
.
14.
Write the piecewise-defined function shown in
each graph.
SOLUTION:
16.
D = {all real numbers} SOLUTION:
The left portion of the graph is the constant function
. g(x) = –8. There is a closed dot at (–6, –8), so the
domain for this part of the function is .
The center portion of the graph is the line g(x) =
0.25x + 2. There are closed dots at (–4, 1) and (4, 3),
15. so the domain for this part is .
The right portion of the graph is the constant function
SOLUTION: g(x) = 4. There is an open circle at (6, 4), so the
constant function is defined for .
Write the piecewise function.
.
Write the piecewise-defined function shown in
each graph.
17.
SOLUTION:
The left portion of the graph is the line g(x) = –x – 4.
16. There is an open circle at (–3, –1), so the domain for
this part of the function is
SOLUTION: .
The left portion of the graph is the constant function The center portion of the graph is the line g(x) = x +
g(x) = –8. There is a closed dot at (–6, –8), so the 1. There are closed dots at (–3, –2) and (1, 2), so the
domain for this part of the function is . domain for this part is .
The center portion of the graph is the line g(x) = The right portion of the graph is the constant function
0.25x + 2. There are closed dots at (–4, 1) and (4, 3), g(x) = –6. There is an open circle at (4, –6), so the
so the domain for this part is . domain for this part is .
Write the piecewise function.
The right portion of the graph is the constant function
g(x) = 4. There is an open circle at (6, 4), so the
constant function is defined for .
Write the piecewise function.
18.
SOLUTION:
The left portion of the graph is the constant function
g(x) = –9. There is an open circle at (–5, –9), so the
17. domain for this part of the function is .
SOLUTION: The center portion of the graph is the line g(x) = x +
The left portion of the graph is the line g(x) = –x – 4. 4. There are closed dots at (0, 4) and (3, 7), so the
There is an open circle at (–3, –1), so the domain for domain for this part is .
this part of the function is
. The right portion of the graph is the line g(x) = x – 3.
The center portion of the graph is the line g(x) = x + There is an open circle at (7, 4), so the domain for
1. There are closed dots at (–3, –2) and (1, 2), so the this part is .
domain for this part is .
Write the piecewise function.
The right portion of the graph is the constant function
g(x) = –6. There is an open circle at (4, –6), so the
domain for this part is .
Write the piecewise function.
19.
SOLUTION:
The left portion of the graph is the constant function
18. g(x) = 8. There is a closed dot at (–1, 8), so the
domain for this part is .
SOLUTION: The center portion of the graph is the line g(x) = 2x.
The left portion of the graph is the constant function There are closed dots at (4, 8) and (6, 12), so the
g(x) = –9. There is an open circle at (–5, –9), so the domain for this part is .
domain for this part of the function is .
The right portion of the graph is the line g(x) = 2x –
15. There is a circle at (7, –1), so the domain for this
The center portion of the graph is the line g(x) = x + part is .
4. There are closed dots at (0, 4) and (3, 7), so the
domain for this part is .
The right portion of the graph is the line g(x) = x – 3. Write the piecewise function.
There is an open circle at (7, 4), so the domain for
this part is .
Write the piecewise function.
Graph each function. Identify the domain and
range.
20.
SOLUTION:
19.
SOLUTION:
The left portion of the graph is the constant function
g(x) = 8. There is a closed dot at (–1, 8), so the D = {all real numbers}
domain for this part is .
R = {all integers}
The center portion of the graph is the line g(x) = 2x.
There are closed dots at (4, 8) and (6, 12), so the
domain for this part is .
21.
The right portion of the graph is the line g(x) = 2x –
15. There is a circle at (7, –1), so the domain for this SOLUTION:
part is .
Write the piecewise function.
D = {all real numbers}
Graph each function. Identify the domain and
range. R = {all integers}
20.
22.
SOLUTION:
SOLUTION:
D = {all real numbers}
D = {all real numbers}
R = {all integers}
R = {all integers}
21.
23.
SOLUTION:
SOLUTION:
D = {all real numbers} The function is defined for all real values of x, so the
domain is all real numbers.
R = {all integers}
D = {all real numbers}
22. The function g(x) is twice of a greatest integer
function. So, g(x) takes only even integer values.
Therefore, the range is R = {all even integers}.
SOLUTION:
Graph each function. Identify the domain and
range.
24.
SOLUTION:
D = {all real numbers}
R = {all integers}
23.
SOLUTION: D = {all real numbers}
25.
SOLUTION:
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
The function g(x) is twice of a greatest integer
function. So, g(x) takes only even integer values.
Therefore, the range is R = {all even integers}.
Graph each function. Identify the domain and D = {all real numbers
}
range.
24.
26.
SOLUTION:
SOLUTION:
D = {all real numbers}
D = {all real numbers}
25.
27.
SOLUTION:
SOLUTION:
D = {all real numbers
} D = {all real numbers}
26.
28.
SOLUTION:
SOLUTION:
D = {all real numbers}
D = {all real numbers}
27.
29.
SOLUTION:
SOLUTION:
D = {all real numbers}
D = {all real numbers}
28.
30. GIVING Patrick is donating money and volunteering
his time to an organization that restores homes for
SOLUTION: the needy. His employer will match his monetary
donations up to $100
a. Identify the type of function that models the total
money received by the charity when Patrick donates
x dollars.
b. Write and graph a function for the situation.
D = {all real numbers}
SOLUTION:
a
. The function is composed of two distinct linear
functions. Therefore, it is a piecewise function.
29. b.
SOLUTION:
D = {all real numbers}
31. CCSS SENSE-MAKINGA car’s speedometer
reads 60 miles an hour.
30. GIVING Patrick is donating money and volunteering a. Write an absolute value function for the difference
his time to an organization that restores homes for between the car’s actual speed a and the reading on
the needy. His employer will match his monetary the speedometer.
donations up to $100
b. What is an appropriate domain for the function?
a. Identify the type of function that models the total Explain your reasoning.
money received by the charity when Patrick donates
x dollars. c. Use the domain to graph the function.
b. Write and graph a function for the situation.
SOLUTION:
a. The absolute value function is .
SOLUTION:
a b. Since the speed of the car cannot be negative, the
. The function is composed of two distinct linear
functions. Therefore, it is a piecewise function. appropriate domain for the function is {a | a ≥ 0}.
c.
b.
32. RECREATIONThe charge for renting a bicycle
from a rental shop for different amounts of time is
shown at the right.
31. CCSS SENSE-MAKINGA car’s speedometer a. Identify the type of function that models this
reads 60 miles an hour. situation.
a. Write an absolute value function for the difference b. Write and graph a function for the situation.
between the car’s actual speed a and the reading on
the speedometer.
b. What is an appropriate domain for the function?
Explain your reasoning.
c. Use the domain to graph the function.
SOLUTION:
a. The absolute value function is .
b. Since the speed of the car cannot be negative, the
appropriate domain for the function is {a | a ≥ 0}.
c.
SOLUTION:
a. The rent is constant in each interval. Therefore,
the situation is best modeled by a step function.
b.
32. RECREATIONThe charge for renting a bicycle
from a rental shop for different amounts of time is
shown at the right.
a. Identify the type of function that models this
situation.
b. Write and graph a function for the situation.
Use each graph to write the absolute value
function.
33.
SOLUTION:
a. The rent is constant in each interval. Therefore,
the situation is best modeled by a step function.
SOLUTION:
b. The graph changes its direction at (0, 0).
The slope of the line in the interval is –0.5.
The slope of the line in the interval is 0.5
.
Therefore, the absolute value function is
.
34.
SOLUTION:
Use each graph to write the absolute value The graph changes its direction at (–5, –4).
function.
The slope of the line in the interval 1.
is –
The slope of the line in the interval
is 1.
Therefore, the absolute value function is
.
33.
Graph each function. Identify the domain and
range.
SOLUTION:
The graph changes its direction at (0, 0).
35.
The slope of the line in the interval is –0.5.
The slope of the line in the interval is 0.5
SOLUTION:
.
Therefore, the absolute value function is
.
D = {all real numbers}
34.
36.
SOLUTION:
The graph changes its direction at (–5, –4).
The slope of the line in the interval 1. SOLUTION:
is –
The slope of the line in the interval
is 1.
Therefore, the absolute value function is
.
Graph each function. Identify the domain and
D = {all real numbers};
range.
R = {all non-negative integers}
35.
37.
SOLUTION:
SOLUTION:
D = {all real numbers}
D = {all real numbers}
36.
.
SOLUTION:
38.
SOLUTION:
D = {all real numbers};
R = {all non-negative integers}
37.
D = {all real numbers}
SOLUTION:
MULTIPLEREPRESENTATIONS
39. Consider the
following absolute value functions.
a. TABULAR
Use a graphing calculator to create a
table of f (x) and g(x) values for x = 4 to x = 4.
–
b. GRAPHICAL
D = {all real numbers} Graph the functions on separate
graphs.
.
c. NUMERICAL
Determine the slope between
each two consecutive points in the table.
d. VERBAL
Describe how the slopes of the two
sections of an absolute value graph are related.
38.
SOLUTION:
a.
SOLUTION:
b.
D = {all real numbers}
c.
MULTIPLEREPRESENTATIONS
39. Consider the
following absolute value functions.
a. TABULAR
Use a graphing calculator to create a d. The two sections of an absolute value graph have
table of f (x) and g(x) values for x = 4 to x = 4.
– opposite slopes. The slope is constant for each
section of the graph.
b. GRAPHICAL
Graph the functions on separate
graphs.
OPENENDED
40. Write an absolute value relation in
c. NUMERICAL
Determine the slope between which the domain is all nonnegative numbers and the
each two consecutive points in the table. range is all real numbers.
d. VERBAL
Describe how the slopes of the two
sections of an absolute value graph are related. SOLUTION:
Sample answer:
| y | = x
SOLUTION:
a.
CHALLENGEGraph
41.
SOLUTION:
b.
CCSS ARGUMENTSFind a counterexample to
42.
c. the following statement and explain your reasoning.
In order to find the greatest integer function of x
when x is not an integer, round x to the nearest
integer.
SOLUTION:
Sample answer: 8.6
d. The two sections of an absolute value graph have The greatest integer function asks for the greatest
opposite slopes. The slope is constant for each integer less than or equal to the given value; thus 8 is
section of the graph. the greatest integer. If we were to round this value to
the nearest integer, we would round up to 9.
OPENENDED
40. Write an absolute value relation in
which the domain is all nonnegative numbers and the
OPENENDED
range is all real numbers. 43. Write an absolute value function in
which f(5) = 3.
–
SOLUTION:
SOLUTION:
Sample answer: |y | = x Sample answer:
CHALLENGEGraph
41. WRITING INMATH
44. Explain how piecewise
functions can be used to accurately represent real-
world problems.
SOLUTION:
SOLUTION:
Sample answer: Piecewise functions can be used to
represent the cost of items when purchased in
quantities, such as a dozen eggs.
SHORT RESPONSEWhat expression gives the
45.
nth term of the linear pattern defined by the table?
CCSS ARGUMENTSFind a counterexample to
42.
the following statement and explain your reasoning.
In order to find the greatest integer function of x
when x is not an integer, round x to the nearest
integer.
SOLUTION:
SOLUTION:
Sample answer:
8.6
The greatest integer function asks for the greatest So, the nth term is 3n + 1.
integer less than or equal to the given value; thus 8 is
the greatest integer. If we were to round this value to
the nearest integer, we would round up to 9. Solve: 5(x + 4) = x + 4
46.
Step 1: 5x + 20 = x + 4
OPENENDED
43. Write an absolute value function in
which f(5) = 3.
– Step 2: 4x + 20 = 4
x = 24
Step 3: 4
SOLUTION:
Sample answer:
x = 6
Step 4:
WRITING INMATH Which is the first incorrect step in the solution
44. Explain how piecewise shown above?
functions can be used to accurately represent real-
world problems.
A Step 4
B Step 3
SOLUTION:
Sample answer: Piecewise functions can be used to
represent the cost of items when purchased in C Step 2
quantities, such as a dozen eggs.
DStep 1
SHORT RESPONSEWhat expression gives the
45.
nth term of the linear pattern defined by the table? SOLUTION:
SOLUTION:
Compare the steps. The first incorrect step in the
solution is on step 3.
Therefore, option B is the correct answer.
So, the nth term is 3n + 1.
NUMBERTHEORY Twelve consecutive integers
Solve: 5(x + 4) = x + 4 47.
46. are arranged in order from least to greatest. If the
sum of the first six integers is 381, what is the sum of
Step 1: 5x + 20 = x + 4 the last six integers?
Step 2: 4x + 20 = 4
F
345
x = 24
Step 3: 4
G381
x = 6
Step 4: H 387
Which is the first incorrect step in the solution
J
shown above? 417
A Step 4
SOLUTION:
Let x be least number in the consecutive integer.
B Step 3
Sum of the first six integers = x + (x + 1) + (x + 2) +
C Step 2 (x + 3) + (x + 4) + (x + 5)
= 6x + 15
D Step 1 Equate 6x + 15 to 381 and solve for x.
SOLUTION:
Therefore, the last 6 integers are 67, 68, 69, 70, 71
and 72.
67 + 68 + 69 + 70 + 71 + 72 = 417
Compare the steps. The first incorrect step in the
solution is on step 3. Therefore, option J is the correct answer.
Therefore, option B is the correct answer.
ACT/SAT For which function does
48.
NUMBERTHEORY Twelve consecutive integers
47.
are arranged in order from least to greatest. If the
sum of the first six integers is 381, what is the sum of
the last six integers?
A
F
345
B
G381
C
H 387
J D
417
E
SOLUTION:
Let x be least number in the consecutive integer.
Sum of the first six integers = x + (x + 1) + (x + 2) + SOLUTION:
(x + 3) + (x + 4) + (x + 5)
= 6x + 15
Equate 6x + 15 to 381 and solve for x.
Therefore, the last 6 integers are 67, 68, 69, 70, 71
and 72.
67 + 68 + 69 + 70 + 71 + 72 = 417
Therefore, option B is the correct answer.
Therefore, option J is the correct answer.
FOOTBALLThe table shows the relationship
49.
ACT/SAT For which function does between the total number of male students per school
48. and the number of students who tried out for the
football team.
a. Find a regression equation for the data.
A b. Determine the correlation coefficient.
B c. Predict how many students will try out for football
at a school with 800 male students.
C
D
E
SOLUTION:
SOLUTION:
a.
y = 0.10x + 30.34
b. r = 0.987
c. Substitute x = 800 in the equation y = 0.10x +
30.34.
Therefore, option B is the correct answer.
FOOTBALL
49. The table shows the relationship
between the total number of male students per school So, at a school with 800 male students, about 110
and the number of students who tried out for the students will try out for football.
football team.
a. Find a regression equation for the data. Write an equation in slope-intercept form for
the line described.
b. Determine the correlation coefficient.
passes through ( 3, 6), perpendicular to y = 2x + 1
50. – – –
c. Predict how many students will try out for football
at a school with 800 male students.
SOLUTION:
The slope of the line y = 2x + 1 is 2.
– –
Therefore, the slope of a line perpendicular to y =
–
2x + 1 is .
Substitute 0.5 for m in the slope-intercept form.
Substitute –3 and –6 for x and y and solve for b
SOLUTION: .
a.
y = 0.10x + 30.34
b. r = 0.987
c. Substitute x = 800 in the equation y = 0.10x + Therefore, the equation of the line which passes
30.34. through the point ( 3, 6) and is perpendicular to y =
– –
2x +1 is y = 0.5x 4.5.
– –
51. passes through (4, 0), parallel to 3x + 2y = 6
So, at a school with 800 male students, about 110 SOLUTION:
students will try out for football. The slope of the line 3x + 2y = 6 is .
Write an equation in slope-intercept form for
Therefore, the slope of a line parallel to the line 3x +
the line described.
2y = 6 is .
passes through ( 3, 6), perpendicular to y = 2x + 1
50. – – –
Substitute m in the slope-intercept form.
for
SOLUTION:
The slope of the line y = 2x + 1 is 2.
– –
Therefore, the slope of a line perpendicular to y =
–
2x + 1 is .
Substitute 4 and 0 for x and y and solve for b.
Substitute 0.5 for m in the slope-intercept form.
Substitute –3 and –6 for x and y and solve for b
.
Therefore, the equation of the line which passes
through the point (4, 0) and is parallel to 3x + 2y = 6
is .
Therefore, the equation of the line which passes
through the point (–3, –6) and is perpendicular to y = passes through the origin, perpendicular to 4x 3y =
2x +1 is y = 0.5x 4.5. 52. –
– – 12
51. passes through (4, 0), parallel to 3x + 2y = 6
SOLUTION:
The slope of the line 4x 3y = 12 is .
–
SOLUTION:
The slope of the line 3x + 2y = 6 is . Therefore, the slope of a line perpendicular to the line
4x 3y .
– = 12 is
Therefore, the slope of a line parallel to the line 3x +
2y = 6 is . Substitute m in the slope-intercept form.
for
Substitute m in the slope-intercept form.
for
Substitute 0 and 0 for x and y and solve for b.
Substitute 4 and 0 for x and y and solve for b.
Therefore, the equation of the line which passes
through the origin and is perpendicular to 4x – 3y =
12 is .
Therefore, the equation of the line which passes
through the point (4, 0) and is parallel to 3x + 2y = 6 Find each value if f (x) = 4x + 6, g(x) = x2, and
– –
is . 2
h(x) = 2x 6x + 9.
– –
f (2c)
53.
passes through the origin, perpendicular to 4x 3y =
52. –
12
SOLUTION:
Substitute 2c for x in the function f (x).
SOLUTION:
The slope of the line 4x 3y = 12 is .
–
Therefore, the slope of a line perpendicular to the line
4x 3y .
– = 12 is g(a + 1)
54.
Substitute m in the slope-intercept form. SOLUTION:
for Substitute a + 1 for x in the function g(x).
Substitute 0 and 0 for x and y and solve for b.
55. h(6)
Therefore, the equation of the line which passes SOLUTION:
through the origin and is perpendicular to 4x 3y = Substitute 6 for x in the function h(x).
–
12 is .
Find each value if f (x) = 4x + 6, g(x) = x2, and
– –
2
h(x) = 2x 6x + 9.
– –
Determine whether the figures below are similar.
56.
f (2c)
53.
SOLUTION:
Substitute 2c for x in the function f (x).
SOLUTION:
The ratio between the length of the rectangles is
.
g(a + 1) The ratio between the width of the rectangles is
54.
.
SOLUTION:
Substitute a + 1 for x in the function g(x).
Since the ratios of the sides are equal, the given
figures are similar.
Graph each equation.
y = 0.25x + 8
57. –
h(6)
55.
SOLUTION:
SOLUTION:
Substitute 6 for x in the function h(x).
56. Determine whether the figures below are similar.
58.
SOLUTION:
SOLUTION:
The ratio between the length of the rectangles is
.
The ratio between the width of the rectangles is
.
Since the ratios of the sides are equal, the given 8x + 4y = 32
figures are similar. 59.
SOLUTION:
Graph each equation.
y = 0.25x + 8
57. –
SOLUTION:
58.
SOLUTION:
59. 8x + 4y = 32
SOLUTION:
Graph each function. Identify the domain and
range.
1.
SOLUTION:
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
The y-coordinates of points on the graph are real
numbers less than or equal to 4, so the range
is .
2.
Graph each function. Identify the domain and
range.
SOLUTION:
1.
SOLUTION:
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
The y-coordinates of points on the graph are real
numbers between 8 and –2 and less than or equal to
The function is defined for all real values of x, so the –8, so the range is .
domain is all real numbers.
Write the piecewise-defined function shown in
D = {all real numbers} each graph.
The y-coordinates of points on the graph are real
numbers less than or equal to 4, so the range
is .
3.
2.
SOLUTION:
The left portion of the graph is the line g(x) = x + 4.
SOLUTION: There is an open circle at (–2, 2), so the domain for
this part of the function is .
The center portion of the graph is the constant
function g(x) = –3. There are closed dots at (–2, –3)
and (3, 3), so the domain for this part is
.
The right portion of the graph is the line g(x) = –2x +
The function is defined for all real values of x, so the 12. There is an open circle at (3, 6), so the domain
domain is all real numbers. for this part is .
D = {all real numbers} Write the piecewise function.
The y-coordinates of points on the graph are real
numbers between 8 and –2 and less than or equal to
–8, so the range is .
Write the piecewise-defined function shown in
each graph.
4.
3.
SOLUTION:
SOLUTION: The left portion of the graph is the constant function
The left portion of the graph is the line g(x) = x + 4. g(x) = 6. There is a closed dot at (–5, 6), so the
There is an open circle at (–2, 2), so the domain for domain for this part is .
this part of the function is .
The center portion of the graph is the line g(x) = –x
The center portion of the graph is the constant + 4. There are open circles at (–5, 9) and (–2, 6), so
function g(x) = –3. There are closed dots at (–2, –3) the domain for this part is .
and (3, 3), so the domain for this part is
The right portion of the graph is the line
.
The right portion of the graph is the line g(x) = –2x + . There is a closed dot at (–2, 0), so
12. There is an open circle at (3, 6), so the domain the domain for this part is .
for this part is .
Write the piecewise function.
Write the piecewise function.
CCSS REASONING
5. Springfield High School’s
theater can hold 250 students. The drama club is
performing a play in the theater. Draw a graph of a
step function that shows the relationship between the
number of tickets sold x and the minimum number of
performances y that the drama club must do.
4.
SOLUTION:
When x is greater than 0 and less than or equal to
250, the drama club needs to do only one
SOLUTION:
The left portion of the graph is the constant function performance. When x is greater than 250 and less
g(x) = 6. There is a closed dot at (–5, 6), so the than or equal to 500, they must do at least two
domain for this part is . performances. Continue the pattern with a table.
The center portion of the graph is the line g(x) = –x
+ 4. There are open circles at (–5, 9) and (–2, 6), so
the domain for this part is .
The right portion of the graph is the line
. There is a closed dot at (–2, 0), so
the domain for this part is .
Write the piecewise function.
2-6 Special Functions
Graph each function. Identify the domain and
CCSS REASONING
5. Springfield High School’s
theater can hold 250 students. The drama club is range.
performing a play in the theater. Draw a graph of a
step function that shows the relationship between the
number of tickets sold x and the minimum number of 6.
performances y that the drama club must do.
SOLUTION:
SOLUTION:
When x is greater than 0 and less than or equal to
250, the drama club needs to do only one
performance. When x is greater than 250 and less
than or equal to 500, they must do at least two
performances. Continue the pattern with a table.
D = {all real numbers}
The function g(x) is a reflection of twice of a
greatest integer function. So, g(x) takes all even
integer values or zero.
R = {all even integers}
7.
SOLUTION:
Graph each function. Identify the domain and
range.
6.
D = {all real numbers}
SOLUTION:
R = {all integers}
Graph each function. Identify the domain and
range.
8.
D = {all real numbers} SOLUTION:
eSolutions Manual - Powered by Cognero Page3
The function g(x) is a reflection of twice of a
greatest integer function. So, g(x) takes all even
integer values or zero.
R = {all even integers}
7. D = {all real numbers}
SOLUTION:
9.
SOLUTION:
D = {all real numbers}
R = {all integers}
Graph each function. Identify the domain and
range. D = {all real numbers}
8.
.
SOLUTION:
10.
SOLUTION:
D = {all real numbers}
D = {all real numbers}
9.
.
SOLUTION:
11.
SOLUTION:
D = {all real numbers}
.
10. D = {all real numbers}
.
SOLUTION:
Graph each function. Identify the domain and
range.
12.
D = {all real numbers} SOLUTION:
.
11.
SOLUTION:
.
.
13.
D = {all real numbers}
.
SOLUTION:
Graph each function. Identify the domain and
range.
12.
SOLUTION: .
.
14.
. SOLUTION:
.
13.
D = {all real numbers}
SOLUTION: .
15.
SOLUTION:
.
.
14.
.
Write the piecewise-defined function shown in
SOLUTION:
each graph.
16.
D = {all real numbers}
. SOLUTION:
The left portion of the graph is the constant function
g(x) = –8. There is a closed dot at (–6, –8), so the
domain for this part of the function is .
15. The center portion of the graph is the line g(x) =
0.25x + 2. There are closed dots at (–4, 1) and (4, 3),
so the domain for this part is .
SOLUTION:
The right portion of the graph is the constant function
g(x) = 4. There is an open circle at (6, 4), so the
constant function is defined for .
Write the piecewise function.
.
Write the piecewise-defined function shown in
each graph.
17.
16.
SOLUTION:
The left portion of the graph is the line g(x) = –x – 4.
SOLUTION: There is an open circle at (–3, –1), so the domain for
The left portion of the graph is the constant function this part of the function is
g(x) = –8. There is a closed dot at (–6, –8), so the .
domain for this part of the function is . The center portion of the graph is the line g(x) = x +
1. There are closed dots at (–3, –2) and (1, 2), so the
The center portion of the graph is the line g(x) = domain for this part is .
0.25x + 2. There are closed dots at (–4, 1) and (4, 3),
so the domain for this part is . The right portion of the graph is the constant function
g(x) = –6. There is an open circle at (4, –6), so the
The right portion of the graph is the constant function domain for this part is .
g(x) = 4. There is an open circle at (6, 4), so the Write the piecewise function.
constant function is defined for .
Write the piecewise function.
18.
SOLUTION:
17. The left portion of the graph is the constant function
g(x) = –9. There is an open circle at (–5, –9), so the
domain for this part of the function is .
SOLUTION:
The left portion of the graph is the line g(x) = –x – 4.
There is an open circle at (–3, –1), so the domain for
this part of the function is The center portion of the graph is the line g(x) = x +
. 4. There are closed dots at (0, 4) and (3, 7), so the
The center portion of the graph is the line g(x) = x + domain for this part is .
1. There are closed dots at (–3, –2) and (1, 2), so the The right portion of the graph is the line g(x) = x – 3.
domain for this part is . There is an open circle at (7, 4), so the domain for
this part is .
The right portion of the graph is the constant function
g(x) = –6. There is an open circle at (4, –6), so the Write the piecewise function.
domain for this part is .
Write the piecewise function.
19.
18.
SOLUTION:
The left portion of the graph is the constant function
g(x) = 8. There is a closed dot at (–1, 8), so the
SOLUTION: domain for this part is .
The left portion of the graph is the constant function
g(x) = –9. There is an open circle at (–5, –9), so the
domain for this part of the function is . The center portion of the graph is the line g(x) = 2x.
There are closed dots at (4, 8) and (6, 12), so the
domain for this part is .
The center portion of the graph is the line g(x) = x +
4. There are closed dots at (0, 4) and (3, 7), so the The right portion of the graph is the line g(x) = 2x –
domain for this part is . 15. There is a circle at (7, –1), so the domain for this
The right portion of the graph is the line g(x) = x – 3. part is .
There is an open circle at (7, 4), so the domain for
this part is . Write the piecewise function.
Write the piecewise function.
Graph each function. Identify the domain and
range.
20.
SOLUTION:
19.
SOLUTION:
The left portion of the graph is the constant function
g(x) = 8. There is a closed dot at (–1, 8), so the
domain for this part is .
D = {all real numbers}
The center portion of the graph is the line g(x) = 2x.
There are closed dots at (4, 8) and (6, 12), so the R = {all integers}
domain for this part is .
The right portion of the graph is the line g(x) = 2x –
15. There is a circle at (7, –1), so the domain for this 21.
part is .
SOLUTION:
Write the piecewise function.
Graph each function. Identify the domain and
range. D = {all real numbers}
20. R = {all integers}
SOLUTION:
22.
SOLUTION:
D = {all real numbers}
R = {all integers}
D = {all real numbers}
R = {all integers}
21.
23.
SOLUTION:
SOLUTION:
D = {all real numbers}
R = {all integers} The function is defined for all real values of x, so the
domain is all real numbers.
22. D = {all real numbers}
The function g(x) is twice of a greatest integer
SOLUTION: function. So, g(x) takes only even integer values.
Therefore, the range is R = {all even integers}.
Graph each function. Identify the domain and
range.
24.
D = {all real numbers}
SOLUTION:
R = {all integers}
23.
SOLUTION:
D = {all real numbers}
25.
The function is defined for all real values of x, so the
domain is all real numbers.
SOLUTION:
D = {all real numbers}
The function g(x) is twice of a greatest integer
function. So, g(x) takes only even integer values.
Therefore, the range is R = {all even integers}.
Graph each function. Identify the domain and
range. D = {all real numbers
}
24.
SOLUTION:
26.
SOLUTION:
D = {all real numbers}
D = {all real numbers}
25.
SOLUTION:
27.
SOLUTION:
D = {all real numbers
}
D = {all real numbers}
26.
SOLUTION:
28.
SOLUTION:
D = {all real numbers}
D = {all real numbers}
27.
SOLUTION:
29.
SOLUTION:
D = {all real numbers}
D = {all real numbers}
28.
SOLUTION:
30. GIVING Patrick is donating money and volunteering
his time to an organization that restores homes for
the needy. His employer will match his monetary
donations up to $100
a. Identify the type of function that models the total
money received by the charity when Patrick donates
x dollars.
D = {all real numbers} b. Write and graph a function for the situation.
SOLUTION:
a
. The function is composed of two distinct linear
functions. Therefore, it is a piecewise function.
29.
b.
SOLUTION:
D = {all real numbers}
31. CCSS SENSE-MAKINGA car’s speedometer
30. GIVING Patrick is donating money and volunteering reads 60 miles an hour.
his time to an organization that restores homes for
the needy. His employer will match his monetary a. Write an absolute value function for the difference
donations up to $100 between the car’s actual speed a and the reading on
the speedometer.
a. Identify the type of function that models the total
money received by the charity when Patrick donates b. What is an appropriate domain for the function?
x dollars. Explain your reasoning.
b. Write and graph a function for the situation. c. Use the domain to graph the function.
SOLUTION:
SOLUTION:
a a. The absolute value function is .
. The function is composed of two distinct linear
functions. Therefore, it is a piecewise function.
b. Since the speed of the car cannot be negative, the
appropriate domain for the function is {a | a ≥ 0}.
b.
c.
32. RECREATIONThe charge for renting a bicycle
from a rental shop for different amounts of time is
31. CCSS SENSE-MAKINGA car’s speedometer
reads 60 miles an hour. shown at the right.
a. Write an absolute value function for the difference a. Identify the type of function that models this
between the car’s actual speed a and the reading on situation.
the speedometer.
b. Write and graph a function for the situation.
b. What is an appropriate domain for the function?
Explain your reasoning.
c. Use the domain to graph the function.
SOLUTION:
a. The absolute value function is .
b. Since the speed of the car cannot be negative, the
appropriate domain for the function is {a | a ≥ 0}.
c.
SOLUTION:
a. The rent is constant in each interval. Therefore,
the situation is best modeled by a step function.
b.
32. RECREATIONThe charge for renting a bicycle
from a rental shop for different amounts of time is
shown at the right.
a. Identify the type of function that models this
situation.
b. Write and graph a function for the situation.
Use each graph to write the absolute value
function.
SOLUTION:
a. The rent is constant in each interval. Therefore,
the situation is best modeled by a step function. 33.
b.
SOLUTION:
The graph changes its direction at (0, 0).
The slope of the line in the interval is –0.5.
The slope of the line in the interval is 0.5
.
Therefore, the absolute value function is
.
34.
Use each graph to write the absolute value
function.
SOLUTION:
The graph changes its direction at (–5, –4).
The slope of the line in the interval 1.
is –
The slope of the line in the interval
is 1.
Therefore, the absolute value function is
33. .
SOLUTION: Graph each function. Identify the domain and
The graph changes its direction at (0, 0).
range.
The slope of the line in the interval is –0.5.
35.
The slope of the line in the interval is 0.5
.
Therefore, the absolute value function is
SOLUTION:
.
D = {all real numbers}
34.
SOLUTION:
The graph changes its direction at (–5, –4).
36.
The slope of the line in the interval 1.
is –
The slope of the line in the interval SOLUTION:
is 1.
Therefore, the absolute value function is
.
Graph each function. Identify the domain and
range.
D = {all real numbers};
35.
R = {all non-negative integers}
SOLUTION:
37.
SOLUTION:
D = {all real numbers}
36.
D = {all real numbers}
SOLUTION: .
38.
SOLUTION:
D = {all real numbers};
R = {all non-negative integers}
37.
D = {all real numbers}
SOLUTION:
MULTIPLEREPRESENTATIONS
39. Consider the
following absolute value functions.
a. TABULAR
Use a graphing calculator to create a
D = {all real numbers} table of f (x) and g(x) values for x = 4 to x = 4.
–
b. GRAPHICAL
. Graph the functions on separate
graphs.
c. NUMERICAL
Determine the slope between
each two consecutive points in the table.
38.
d. VERBAL
Describe how the slopes of the two
sections of an absolute value graph are related.
SOLUTION:
SOLUTION:
a.
b.
D = {all real numbers}
MULTIPLEREPRESENTATIONS
39. Consider the
following absolute value functions. c.
a. TABULAR
Use a graphing calculator to create a
table of f (x) and g(x) values for x = 4 to x = 4.
–
d.
b. GRAPHICAL The two sections of an absolute value graph have
Graph the functions on separate opposite slopes. The slope is constant for each
graphs. section of the graph.
c. NUMERICAL
Determine the slope between
each two consecutive points in the table.
OPENENDED
40. Write an absolute value relation in
which the domain is all nonnegative numbers and the
d. VERBAL
Describe how the slopes of the two range is all real numbers.
sections of an absolute value graph are related.
SOLUTION:
SOLUTION: Sample answer:
a. | y | = x
CHALLENGEGraph
41.
SOLUTION:
b.
c.
CCSS ARGUMENTSFind a counterexample to
42.
the following statement and explain your reasoning.
In order to find the greatest integer function of x
when x is not an integer, round x to the nearest
integer.
d. The two sections of an absolute value graph have
opposite slopes. The slope is constant for each SOLUTION:
section of the graph. Sample answer: 8.6
The greatest integer function asks for the greatest
integer less than or equal to the given value; thus 8 is
the greatest integer. If we were to round this value to
OPENENDED
40. Write an absolute value relation in the nearest integer, we would round up to 9.
which the domain is all nonnegative numbers and the
range is all real numbers.
OPENENDED
43. Write an absolute value function in
which f(5) = 3.
–
SOLUTION:
Sample answer:
| y | = x
SOLUTION:
Sample answer:
CHALLENGEGraph
41.
WRITING INMATH
44. Explain how piecewise
SOLUTION: functions can be used to accurately represent real-
world problems.
SOLUTION:
Sample answer: Piecewise functions can be used to
represent the cost of items when purchased in
quantities, such as a dozen eggs.
SHORT RESPONSEWhat expression gives the
45.
CCSS ARGUMENTSFind a counterexample to
42. nth term of the linear pattern defined by the table?
the following statement and explain your reasoning.
In order to find the greatest integer function of x
when x is not an integer, round x to the nearest
integer.
SOLUTION:
SOLUTION:
Sample answer: 8.6
The greatest integer function asks for the greatest
integer less than or equal to the given value; thus 8 is
the greatest integer. If we were to round this value to
the nearest integer, we would round up to 9. So, the nth term is 3n + 1.
46. Solve: 5(x + 4) = x + 4
OPENENDED
43. Write an absolute value function in
which f(5) = 3.
– Step 1: 5x + 20 = x + 4
Step 2: 4x + 20 = 4
SOLUTION:
Sample answer:
x = 24
Step 3: 4
x = 6
WRITING INMATH Step 4:
44. Explain how piecewise
functions can be used to accurately represent real-
world problems. Which is the first incorrect step in the solution
shown above?
A
SOLUTION: Step 4
Sample answer: Piecewise functions can be used to
represent the cost of items when purchased in B Step 3
quantities, such as a dozen eggs.
C Step 2
SHORT RESPONSEWhat expression gives the
45. D Step 1
nth term of the linear pattern defined by the table?
SOLUTION:
SOLUTION:
So, the nth term is 3n + 1. Compare the steps. The first incorrect step in the
solution is on step 3.
Therefore, option B is the correct answer.
46. Solve: 5(x + 4) = x + 4
NUMBERTHEORY Twelve consecutive integers
Step 1: 5x + 20 = x + 4 47.
are arranged in order from least to greatest. If the
sum of the first six integers is 381, what is the sum of
Step 2: 4x + 20 = 4
the last six integers?
x = 24
Step 3: 4 F
345
x = 6
Step 4:
G381
Which is the first incorrect step in the solution
shown above? H 387
A Step 4 J
417
B Step 3
SOLUTION:
Let x be least number in the consecutive integer.
C Step 2
Sum of the first six integers = x + (x + 1) + (x + 2) +
D Step 1 (x + 3) + (x + 4) + (x + 5)
= 6x + 15
Equate 6x + 15 to 381 and solve for x.
SOLUTION:
Therefore, the last 6 integers are 67, 68, 69, 70, 71
and 72.
Compare the steps. The first incorrect step in the
solution is on step 3. 67 + 68 + 69 + 70 + 71 + 72 = 417
Therefore, option B is the correct answer.
Therefore, option J is the correct answer.
NUMBERTHEORY Twelve consecutive integers
47. ACT/SAT
are arranged in order from least to greatest. If the 48. For which function does
sum of the first six integers is 381, what is the sum of
the last six integers?
F
345
A
G381
B
H 387
J C
417
D
SOLUTION:
Let x be least number in the consecutive integer.
E
Sum of the first six integers = x + (x + 1) + (x + 2) +
(x + 3) + (x + 4) + (x + 5)
= 6x + 15 SOLUTION:
Equate 6x + 15 to 381 and solve for x.
Therefore, the last 6 integers are 67, 68, 69, 70, 71
and 72.
67 + 68 + 69 + 70 + 71 + 72 = 417
Therefore, option J is the correct answer.
Therefore, option B is the correct answer.
ACT/SAT For which function does
48.
FOOTBALLThe table shows the relationship
49.
between the total number of male students per school
and the number of students who tried out for the
football team.
A
a. Find a regression equation for the data.
B b. Determine the correlation coefficient.
C c. Predict how many students will try out for football
at a school with 800 male students.
D
E
SOLUTION:
SOLUTION:
a.
y = 0.10x + 30.34
b. r = 0.987
c. Substitute x = 800 in the equation y = 0.10x +
30.34.
Therefore, option B is the correct answer.
FOOTBALLThe table shows the relationship
49.
between the total number of male students per school
and the number of students who tried out for the
football team.
So, at a school with 800 male students, about 110
students will try out for football.
a. Find a regression equation for the data.
b. Determine the correlation coefficient. Write an equation in slope-intercept form for
the line described.
c. Predict how many students will try out for football passes through ( 3, 6), perpendicular to y = 2x + 1
at a school with 800 male students. 50. – – –
SOLUTION:
The slope of the line y = 2x + 1 is 2.
– –
Therefore, the slope of a line perpendicular to y =
–
2x + 1 is .
Substitute 0.5 for m in the slope-intercept form.
SOLUTION:
a.
y = 0.10x + 30.34 Substitute 3 and 6 for x and y and solve for b
– –
.
b. r = 0.987
c. Substitute x = 800 in the equation y = 0.10x +
30.34.
Therefore, the equation of the line which passes
through the point ( 3, 6) and is perpendicular to y =
– –
–2x +1 is y = 0.5x – 4.5.
passes through (4, 0), parallel to 3x + 2y = 6
So, at a school with 800 male students, about 110 51.
students will try out for football.
SOLUTION:
Write an equation in slope-intercept form for The slope of the line 3x + 2y = 6 is .
the line described.
Therefore, the slope of a line parallel to the line 3x +
passes through ( 3, 6), perpendicular to y = 2x + 1
50. – – –
2y = 6 is .
SOLUTION:
The slope of the line y = 2x + 1 is 2.
– – Substitute m in the slope-intercept form.
for
Therefore, the slope of a line perpendicular to y =
–
2x + 1 is .
Substitute 0.5 for m in the slope-intercept form.
Substitute 4 and 0 for x and y and solve for b.
Substitute –3 and –6 for x and y and solve for b
.
Therefore, the equation of the line which passes
through the point (4, 0) and is parallel to 3x + 2y = 6
Therefore, the equation of the line which passes is .
through the point (–3, –6) and is perpendicular to y =
2x +1 is y = 0.5x 4.5.
– –
52. passes through the origin, perpendicular to 4x – 3y =
passes through (4, 0), parallel to 3x + 2y = 6 12
51.
SOLUTION:
SOLUTION:
The slope of the line 4x 3y = 12 is .
The slope of the line 3x + 2y = 6 is . –
Therefore, the slope of a line perpendicular to the line
Therefore, the slope of a line parallel to the line 3x +
4x 3y .
2y = 6 is . – = 12 is
Substitute m in the slope-intercept form.
Substitute m in the slope-intercept form. for
for
Substitute 0 and 0 for x and y and solve for b.
Substitute 4 and 0 for x and y and solve for b.
Therefore, the equation of the line which passes
through the origin and is perpendicular to 4x – 3y =
Therefore, the equation of the line which passes 12 is .
through the point (4, 0) and is parallel to 3x + 2y = 6
is .
Find each value if f (x) = 4x + 6, g(x) = x2, and
– –
2
h(x) = 2x 6x + 9.
– –
52. passes through the origin, perpendicular to 4x – 3y =
12 f (2c)
53.
SOLUTION:
SOLUTION:
The slope of the line 4x 3y = 12 is . Substitute 2c for x in the function f (x).
–
Therefore, the slope of a line perpendicular to the line
4x – 3y = 12 is .
g(a + 1)
54.
Substitute m in the slope-intercept form.
for
SOLUTION:
Substitute a + 1 for x in the function g(x).
Substitute 0 and 0 for x and y and solve for b.
h(6)
Therefore, the equation of the line which passes 55.
through the origin and is perpendicular to 4x 3y =
–
12 is . SOLUTION:
Substitute 6 for x in the function h(x).
Find each value if f (x) = 4x + 6, g(x) = x2, and
– –
2
h(x) = 2x 6x + 9.
– –
f (2c)
53.
Determine whether the figures below are similar.
56.
SOLUTION:
Substitute 2c for x in the function f (x).
SOLUTION:
The ratio between the length of the rectangles is
.
54. g(a + 1)
The ratio between the width of the rectangles is
SOLUTION:
Substitute a + 1 for x in the function g(x). .
Since the ratios of the sides are equal, the given
figures are similar.
Graph each equation.
h(6)
55.
y = 0.25x + 8
57. –
SOLUTION:
Substitute 6 for x in the function h(x). SOLUTION:
56. Determine whether the figures below are similar.
58.
SOLUTION:
The ratio between the length of the rectangles is SOLUTION:
.
The ratio between the width of the rectangles is
.
Since the ratios of the sides are equal, the given
figures are similar.
8x + 4y = 32
59.
Graph each equation.
SOLUTION:
y = 0.25x + 8
57. –
SOLUTION:
58.
SOLUTION:
59. 8x + 4y = 32
SOLUTION:
Graph each function. Identify the domain and
range.
1.
SOLUTION:
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
The y-coordinates of points on the graph are real
numbers less than or equal to 4, so the range
is .
2.
SOLUTION:
Graph each function. Identify the domain and
range.
The function is defined for all real values of x, so the
1. domain is all real numbers.
D = {all real numbers}
SOLUTION: The y-coordinates of points on the graph are real
numbers between 8 and –2 and less than or equal to
–8, so the range is .
Write the piecewise-defined function shown in
each graph.
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
The y-coordinates of points on the graph are real 3.
numbers less than or equal to 4, so the range
is . SOLUTION:
The left portion of the graph is the line g(x) = x + 4.
There is an open circle at (–2, 2), so the domain for
this part of the function is .
2. The center portion of the graph is the constant
function g(x) = –3. There are closed dots at (–2, –3)
and (3, 3), so the domain for this part is
SOLUTION: .
The right portion of the graph is the line g(x) = –2x +
12. There is an open circle at (3, 6), so the domain
for this part is .
Write the piecewise function.
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
The y-coordinates of points on the graph are real
numbers between 8 and –2 and less than or equal to
–8, so the range is .
Write the piecewise-defined function shown in
each graph.
4.
SOLUTION:
The left portion of the graph is the constant function
g(x) = 6. There is a closed dot at (–5, 6), so the
domain for this part is .
The center portion of the graph is the line g(x) = –x
3. + 4. There are open circles at (–5, 9) and (–2, 6), so
the domain for this part is .
SOLUTION:
The left portion of the graph is the line g(x) = x + 4. The right portion of the graph is the line
There is an open circle at (–2, 2), so the domain for
this part of the function is . . There is a closed dot at (–2, 0), so
the domain for this part is .
The center portion of the graph is the constant Write the piecewise function.
function g(x) = –3. There are closed dots at (–2, –3)
and (3, 3), so the domain for this part is
.
The right portion of the graph is the line g(x) = –2x +
12. There is an open circle at (3, 6), so the domain
for this part is .
Write the piecewise function.
CCSS REASONING
5. Springfield High School’s
theater can hold 250 students. The drama club is
performing a play in the theater. Draw a graph of a
step function that shows the relationship between the
number of tickets sold x and the minimum number of
performances y that the drama club must do.
SOLUTION:
When x is greater than 0 and less than or equal to
250, the drama club needs to do only one
performance. When x is greater than 250 and less
than or equal to 500, they must do at least two
performances. Continue the pattern with a table.
4.
SOLUTION:
The left portion of the graph is the constant function
g(x) = 6. There is a closed dot at (–5, 6), so the
domain for this part is .
The center portion of the graph is the line g(x) = –x
+ 4. There are open circles at (–5, 9) and (–2, 6), so
the domain for this part is .
The right portion of the graph is the line
. There is a closed dot at (–2, 0), so
the domain for this part is .
Write the piecewise function.
Graph each function. Identify the domain and
range.
6.
SOLUTION:
CCSS REASONING
5. Springfield High School’s
theater can hold 250 students. The drama club is
performing a play in the theater. Draw a graph of a
step function that shows the relationship between the
number of tickets sold x and the minimum number of
performances y that the drama club must do.
SOLUTION:
When x is greater than 0 and less than or equal to D = {all real numbers}
250, the drama club needs to do only one
performance. When x is greater than 250 and less The function g(x) is a reflection of twice of a
than or equal to 500, they must do at least two greatest integer function. So, g(x) takes all even
performances. Continue the pattern with a table. integer values or zero.
R = {all even integers}
7.
SOLUTION:
D = {all real numbers}
Graph each function. Identify the domain and R = {all integers}
range.
Graph each function. Identify the domain and
6.
range.
8.
SOLUTION:
SOLUTION:
D = {all real numbers}
The function g(x) is a reflection of twice of a D = {all real numbers}
greatest integer function. So, g(x) takes all even
integer values or zero.
R = {all even integers}
9.
7.
SOLUTION:
SOLUTION:
D = {all real numbers} D = {all real numbers}
R = {all integers} .
2-6 Special Functions
Graph each function. Identify the domain and
10.
range.
8.
SOLUTION:
SOLUTION:
D = {all real numbers}
D = {all real numbers} .
11.
9.
SOLUTION:
SOLUTION:
D = {all real numbers}
D = {all real numbers}
.
.
Graph each function. Identify the domain and
range.
10.
12.
SOLUTION:
SOLUTION:
eSolutions Manual - Powered by Cognero Page4
D = {all real numbers}
.
.
.
11.
SOLUTION:
13.
SOLUTION:
D = {all real numbers}
.
Graph each function. Identify the domain and
range. .
.
12.
14.
SOLUTION:
SOLUTION:
.
. D = {all real numbers}
.
13.
15.
SOLUTION:
SOLUTION:
. .
. Write the piecewise-defined function shown in
each graph.
14.
SOLUTION:
16.
SOLUTION:
The left portion of the graph is the constant function
g(x) = –8. There is a closed dot at (–6, –8), so the
domain for this part of the function is .
D = {all real numbers} The center portion of the graph is the line g(x) =
0.25x + 2. There are closed dots at (–4, 1) and (4, 3),
. so the domain for this part is .
The right portion of the graph is the constant function
g(x) = 4. There is an open circle at (6, 4), so the
constant function is defined for .
15.
Write the piecewise function.
SOLUTION:
.
Write the piecewise-defined function shown in
each graph. 17.
SOLUTION:
The left portion of the graph is the line g(x) = –x – 4.
There is an open circle at (–3, –1), so the domain for
this part of the function is
.
The center portion of the graph is the line g(x) = x +
1. There are closed dots at (–3, –2) and (1, 2), so the
16.
domain for this part is .
SOLUTION: The right portion of the graph is the constant function
The left portion of the graph is the constant function g(x) = –6. There is an open circle at (4, –6), so the
g(x) = –8. There is a closed dot at (–6, –8), so the domain for this part is .
domain for this part of the function is .
Write the piecewise function.
The center portion of the graph is the line g(x) =
0.25x + 2. There are closed dots at (–4, 1) and (4, 3),
so the domain for this part is .
The right portion of the graph is the constant function
g(x) = 4. There is an open circle at (6, 4), so the
constant function is defined for .
Write the piecewise function.
18.
SOLUTION:
The left portion of the graph is the constant function
g(x) = –9. There is an open circle at (–5, –9), so the
domain for this part of the function is .
The center portion of the graph is the line g(x) = x +
4. There are closed dots at (0, 4) and (3, 7), so the
17. domain for this part is .
The right portion of the graph is the line g(x) = x – 3.
There is an open circle at (7, 4), so the domain for
SOLUTION:
The left portion of the graph is the line g(x) = –x – 4. this part is .
There is an open circle at (–3, –1), so the domain for
this part of the function is Write the piecewise function.
.
The center portion of the graph is the line g(x) = x +
1. There are closed dots at (–3, –2) and (1, 2), so the
domain for this part is .
The right portion of the graph is the constant function
g(x) = –6. There is an open circle at (4, –6), so the
domain for this part is .
Write the piecewise function.
19.
SOLUTION:
The left portion of the graph is the constant function
g(x) = 8. There is a closed dot at (–1, 8), so the
domain for this part is .
The center portion of the graph is the line g(x) = 2x.
There are closed dots at (4, 8) and (6, 12), so the
18. domain for this part is .
SOLUTION: The right portion of the graph is the line g(x) = 2x –
The left portion of the graph is the constant function 15. There is a circle at (7, –1), so the domain for this
g(x) = –9. There is an open circle at (–5, –9), so the part is .
domain for this part of the function is .
Write the piecewise function.
The center portion of the graph is the line g(x) = x +
4. There are closed dots at (0, 4) and (3, 7), so the
domain for this part is .
The right portion of the graph is the line g(x) = x – 3.
There is an open circle at (7, 4), so the domain for
this part is . Graph each function. Identify the domain and
range.
Write the piecewise function.
20.
SOLUTION:
D = {all real numbers}
19.
R = {all integers}
SOLUTION:
The left portion of the graph is the constant function
g(x) = 8. There is a closed dot at (–1, 8), so the
domain for this part is . 21.
The center portion of the graph is the line g(x) = 2x.
SOLUTION:
There are closed dots at (4, 8) and (6, 12), so the
domain for this part is .
The right portion of the graph is the line g(x) = 2x –
15. There is a circle at (7, –1), so the domain for this
part is .
Write the piecewise function.
D = {all real numbers}
R = {all integers}
22.
Graph each function. Identify the domain and
range.
SOLUTION:
20.
SOLUTION:
D = {all real numbers}
R = {all integers}
D = {all real numbers}
23.
R = {all integers}
SOLUTION:
21.
SOLUTION:
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
D = {all real numbers} The function g(x) is twice of a greatest integer
function. So, g(x) takes only even integer values.
R = {all integers} Therefore, the range is R = {all even integers}.
Graph each function. Identify the domain and
22.
range.
SOLUTION:
24.
SOLUTION:
D = {all real numbers}
R = {all integers}
D = {all real numbers}
23.
SOLUTION:
25.
SOLUTION:
The function is defined for all real values of x, so the
domain is all real numbers.
D = {all real numbers}
D = {all real numbers
The function g(x) is twice of a greatest integer
}
function. So, g(x) takes only even integer values.
Therefore, the range is R = {all even integers}.
Graph each function. Identify the domain and 26.
range.
SOLUTION:
24.
SOLUTION:
D = {all real numbers}
D = {all real numbers}
27.
SOLUTION:
25.
SOLUTION:
D = {all real numbers}
D = {all real numbers
}
28.
SOLUTION:
26.
SOLUTION:
D = {all real numbers}
D = {all real numbers}
29.
SOLUTION:
27.
SOLUTION:
D = {all real numbers}
D = {all real numbers}
30. GIVING Patrick is donating money and volunteering
his time to an organization that restores homes for
the needy. His employer will match his monetary
donations up to $100
28.
a. Identify the type of function that models the total
money received by the charity when Patrick donates
SOLUTION: x dollars.
b. Write and graph a function for the situation.
SOLUTION:
a
. The function is composed of two distinct linear
functions. Therefore, it is a piecewise function.
D = {all real numbers}
b.
29.
SOLUTION:
31. CCSS SENSE-MAKINGA car’s speedometer
reads 60 miles an hour.
D = {all real numbers} a. Write an absolute value function for the difference
between the car’s actual speed a and the reading on
the speedometer.
b. What is an appropriate domain for the function?
30. GIVING Patrick is donating money and volunteering Explain your reasoning.
his time to an organization that restores homes for
the needy. His employer will match his monetary c. Use the domain to graph the function.
donations up to $100
SOLUTION:
a. Identify the type of function that models the total a. The absolute value function is .
money received by the charity when Patrick donates
x dollars. b. Since the speed of the car cannot be negative, the
appropriate domain for the function is {a | a ≥ 0}.
b. Write and graph a function for the situation.
c.
SOLUTION:
a
. The function is composed of two distinct linear
functions. Therefore, it is a piecewise function.
b.
32. RECREATIONThe charge for renting a bicycle
from a rental shop for different amounts of time is
shown at the right.
a. Identify the type of function that models this
situation.
b. Write and graph a function for the situation.
31. CCSS SENSE-MAKINGA car’s speedometer
reads 60 miles an hour.
a. Write an absolute value function for the difference
between the car’s actual speed a and the reading on
the speedometer.
b. What is an appropriate domain for the function?
Explain your reasoning.
c. Use the domain to graph the function.
SOLUTION:
a. The absolute value function is .
SOLUTION:
b. Since the speed of the car cannot be negative, the a. The rent is constant in each interval. Therefore,
appropriate domain for the function is {a | a ≥ 0}. the situation is best modeled by a step function.
b.
c.
32. RECREATIONThe charge for renting a bicycle
from a rental shop for different amounts of time is
shown at the right.
a. Identify the type of function that models this
situation.
b. Write and graph a function for the situation.
Use each graph to write the absolute value
function.
33.
SOLUTION:
The graph changes its direction at (0, 0).
SOLUTION: The slope of the line in the interval is –0.5.
a. The rent is constant in each interval. Therefore,
the situation is best modeled by a step function.
The slope of the line in the interval is 0.5
b. .
Therefore, the absolute value function is
.
34.
SOLUTION:
The graph changes its direction at (–5, –4).
The slope of the line in the interval 1.
is –
The slope of the line in the interval
Use each graph to write the absolute value is 1.
function.
Therefore, the absolute value function is
.
Graph each function. Identify the domain and
range.
35.
33.
SOLUTION:
The graph changes its direction at (0, 0). SOLUTION:
The slope of the line in the interval is –0.5.
The slope of the line in the interval is 0.5
.
Therefore, the absolute value function is
.
D = {all real numbers}
36.
34.
SOLUTION:
SOLUTION:
The graph changes its direction at (–5, –4).
The slope of the line in the interval 1.
is –
The slope of the line in the interval
is 1.
Therefore, the absolute value function is D = {all real numbers};
.
R = {all non-negative integers}
Graph each function. Identify the domain and
range.
37.
35.
SOLUTION:
SOLUTION:
D = {all real numbers} D = {all real numbers}
.
36.
38.
SOLUTION:
SOLUTION:
D = {all real numbers};
R = {all non-negative integers}
D = {all real numbers}
37.
MULTIPLEREPRESENTATIONS
39. Consider the
following absolute value functions.
SOLUTION:
a. TABULAR
Use a graphing calculator to create a
table of f (x) and g(x) values for x = 4 to x = 4.
–
b. GRAPHICAL
Graph the functions on separate
graphs.
c. NUMERICAL
Determine the slope between
each two consecutive points in the table.
D = {all real numbers}
d. VERBAL
Describe how the slopes of the two
. sections of an absolute value graph are related.
SOLUTION:
a.
38.
SOLUTION: b.
c.
D = {all real numbers}
MULTIPLEREPRESENTATIONS
39. Consider the
following absolute value functions. d. The two sections of an absolute value graph have
opposite slopes. The slope is constant for each
section of the graph.
a. TABULAR
Use a graphing calculator to create a
OPENENDED
table of f (x) and g(x) values for x = 4 to x = 4. 40. Write an absolute value relation in
– which the domain is all nonnegative numbers and the
range is all real numbers.
b. GRAPHICAL
Graph the functions on separate
graphs.
SOLUTION:
Sample answer:
| y | = x
c. NUMERICAL
Determine the slope between
each two consecutive points in the table.
CHALLENGEGraph
d. VERBAL 41.
Describe how the slopes of the two
sections of an absolute value graph are related.
SOLUTION:
SOLUTION:
a.
b.
CCSS ARGUMENTSFind a counterexample to
42.
the following statement and explain your reasoning.
In order to find the greatest integer function of x
when x is not an integer, round x to the nearest
integer.
c.
SOLUTION:
Sample answer: 8.6
The greatest integer function asks for the greatest
integer less than or equal to the given value; thus 8 is
the greatest integer. If we were to round this value to
the nearest integer, we would round up to 9.
d. The two sections of an absolute value graph have
OPENENDED
opposite slopes. The slope is constant for each 43. Write an absolute value function in
section of the graph. which f(5) = 3.
–
SOLUTION:
OPENENDED
40. Write an absolute value relation in
which the domain is all nonnegative numbers and the Sample answer:
range is all real numbers.
WRITING INMATH
44. Explain how piecewise
SOLUTION: functions can be used to accurately represent real-
Sample answer:
| y | = x world problems.
SOLUTION:
CHALLENGEGraph
41.
Sample answer: Piecewise functions can be used to
represent the cost of items when purchased in
SOLUTION: quantities, such as a dozen eggs.
SHORT RESPONSEWhat expression gives the
45.
nth term of the linear pattern defined by the table?
SOLUTION:
CCSS ARGUMENTSFind a counterexample to
42.
the following statement and explain your reasoning.
In order to find the greatest integer function of x
when x is not an integer, round x to the nearest
So, the nth term is 3n + 1.
integer.
46. Solve: 5(x + 4) = x + 4
SOLUTION:
Sample answer: 8.6
The greatest integer function asks for the greatest Step 1: 5x + 20 = x + 4
integer less than or equal to the given value; thus 8 is
the greatest integer. If we were to round this value to Step 2: 4x + 20 = 4
the nearest integer, we would round up to 9.
x = 24
Step 3: 4
OPENENDED
43. Write an absolute value function in x = 6
Step 4:
which f(5) = –3.
Which is the first incorrect step in the solution
shown above?
SOLUTION:
Sample answer:
A Step 4
WRITING INMATH B Step 3
44. Explain how piecewise
functions can be used to accurately represent real-
world problems. C Step 2
D Step 1
SOLUTION:
Sample answer: Piecewise functions can be used to
represent the cost of items when purchased in SOLUTION:
quantities, such as a dozen eggs.
SHORT RESPONSEWhat expression gives the
45.
nth term of the linear pattern defined by the table?
Compare the steps. The first incorrect step in the
solution is on step 3.
Therefore, option B is the correct answer.
SOLUTION:
NUMBERTHEORY Twelve consecutive integers
47.
are arranged in order from least to greatest. If the
sum of the first six integers is 381, what is the sum of
So, the nth term is 3n + 1. the last six integers?
F
Solve: 5(x + 4) = x + 4 345
46.
Step 1: 5x + 20 = x + 4 G381
H 387
Step 2: 4x + 20 = 4
J
x = 24 417
Step 3: 4
x = 6
Step 4:
SOLUTION:
Let x be least number in the consecutive integer.
Which is the first incorrect step in the solution
shown above?
Sum of the first six integers = x + (x + 1) + (x + 2) +
(x + 3) + (x + 4) + (x + 5)
A Step 4 = 6x + 15
Equate 6x + 15 to 381 and solve for x.
B Step 3
C Step 2
D Step 1
Therefore, the last 6 integers are 67, 68, 69, 70, 71
and 72.
SOLUTION:
67 + 68 + 69 + 70 + 71 + 72 = 417
Therefore, option J is the correct answer.
ACT/SAT For which function does
48.
Compare the steps. The first incorrect step in the
solution is on step 3.
Therefore, option B is the correct answer.
A
NUMBERTHEORY Twelve consecutive integers
47.
are arranged in order from least to greatest. If the
sum of the first six integers is 381, what is the sum of B
the last six integers?
C
F
345
D
G381
H 387 E
J
417
SOLUTION:
SOLUTION:
Let x be least number in the consecutive integer.
Sum of the first six integers = x + (x + 1) + (x + 2) +
(x + 3) + (x + 4) + (x + 5)
= 6x + 15
Equate 6x + 15 to 381 and solve for x.
Therefore, option B is the correct answer.
Therefore, the last 6 integers are 67, 68, 69, 70, 71
and 72.
FOOTBALLThe table shows the relationship
49.
between the total number of male students per school
67 + 68 + 69 + 70 + 71 + 72 = 417 and the number of students who tried out for the
football team.
Therefore, option J is the correct answer.
a. Find a regression equation for the data.
ACT/SAT For which function does
48. b. Determine the correlation coefficient.
c. Predict how many students will try out for football
at a school with 800 male students.
A
B
C
D
E
SOLUTION:
a.
y = 0.10x + 30.34
SOLUTION:
b. r = 0.987
c. Substitute x = 800 in the equation y = 0.10x +
30.34.
So, at a school with 800 male students, about 110
students will try out for football.
Therefore, option B is the correct answer.
Write an equation in slope-intercept form for
FOOTBALLThe table shows the relationship
49.
between the total number of male students per school the line described.
and the number of students who tried out for the
passes through ( 3, 6), perpendicular to y = 2x + 1
football team. 50. – – –
a. Find a regression equation for the data.
SOLUTION:
The slope of the line y = 2x + 1 is 2.
b. Determine the correlation coefficient. – –
Therefore, the slope of a line perpendicular to y =
–
c. Predict how many students will try out for football 2x + 1 is .
at a school with 800 male students.
Substitute 0.5 for m in the slope-intercept form.
Substitute –3 and –6 for x and y and solve for b
.
SOLUTION:
a. Therefore, the equation of the line which passes
y = 0.10x + 30.34 through the point ( 3, 6) and is perpendicular to y =
– –
2x +1 is y = 0.5x 4.5.
b. r = 0.987 – –
c. Substitute x = 800 in the equation y = 0.10x + passes through (4, 0), parallel to 3x + 2y = 6
30.34. 51.
SOLUTION:
The slope of the line 3x + 2y = 6 is .
Therefore, the slope of a line parallel to the line 3x +
So, at a school with 800 male students, about 110 2y = 6 is .
students will try out for football.
Write an equation in slope-intercept form for Substitute m in the slope-intercept form.
for
the line described.
passes through ( 3, 6), perpendicular to y = 2x + 1
50. – – –
SOLUTION: Substitute 4 and 0 for x and y and solve for b.
The slope of the line y = 2x + 1 is 2.
– –
Therefore, the slope of a line perpendicular to y =
–
2x + 1 is .
Substitute 0.5 for m in the slope-intercept form.
Therefore, the equation of the line which passes
through the point (4, 0) and is parallel to 3x + 2y
= 6
is .
Substitute –3 and –6 for x and y and solve for b
.
52. passes through the origin, perpendicular to 4x – 3y =
12
Therefore, the equation of the line which passes
through the point ( 3, 6) and is perpendicular to y = SOLUTION:
– –
–2x +1 is y = 0.5x – 4.5. The slope of the line 4x 3y = 12 is .
–
passes through (4, 0), parallel to 3x + 2y = 6 Therefore, the slope of a line perpendicular to the line
51.
4x – 3y = 12 is .
SOLUTION:
Substitute m in the slope-intercept form.
The slope of the line 3x + 2y = 6 is . for
Therefore, the slope of a line parallel to the line 3x +
2y = 6 is .
Substitute 0 and 0 for x and y and solve for b.
Substitute m in the slope-intercept form.
for
Therefore, the equation of the line which passes
through the origin and is perpendicular to 4x 3y =
Substitute 4 and 0 for x and y and solve for b. –
12 is .
Find each value if f (x) = 4x + 6, g(x) = x2, and
– –
2
h(x) = 2x 6x + 9.
– –
f (2c)
Therefore, the equation of the line which passes 53.
through the point (4, 0) and is parallel to 3x + 2y
= 6
is . SOLUTION:
Substitute 2c for x in the function f (x).
passes through the origin, perpendicular to 4x 3y =
52. –
12
SOLUTION:
The slope of the line 4x 3y = 12 is . 54. g(a + 1)
–
Therefore, the slope of a line perpendicular to the line SOLUTION:
Substitute a + 1 for x in the function g(x).
4x 3y .
– = 12 is
Substitute m in the slope-intercept form.
for
55. h(6)
Substitute 0 and 0 for x and y and solve for b.
SOLUTION:
Substitute 6 for x in the function h(x).
Therefore, the equation of the line which passes
through the origin and is perpendicular to 4x – 3y =
12 is .
56. Determine whether the figures below are similar.
Find each value if f (x) = 4x + 6, g(x) = x2, and
– –
2
h(x) = 2x 6x + 9.
– –
f (2c)
53.
SOLUTION:
The ratio between the length of the rectangles is
SOLUTION: .
Substitute 2c for x in the function f (x).
The ratio between the width of the rectangles is
.
Since the ratios of the sides are equal, the given
g(a + 1) figures are similar.
54.
SOLUTION:
Substitute a + 1 for x in the function g(x). Graph each equation.
y = 0.25x + 8
57. –
SOLUTION:
55. h(6)
SOLUTION:
Substitute 6 for x in the function h(x).
58.
Determine whether the figures below are similar. SOLUTION:
56.
SOLUTION:
The ratio between the length of the rectangles is
.
8x + 4y = 32
The ratio between the width of the rectangles is 59.
.
SOLUTION:
Since the ratios of the sides are equal, the given
figures are similar.
Graph each equation.
y = 0.25x + 8
57. –
SOLUTION:
58.
SOLUTION:
59. 8x + 4y = 32
SOLUTION:
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