2-6 Special Functions
Graph each function. Identify the domain and range.
13. SOLUTION: .
.
15. SOLUTION: .
Write the piecewise-defined function shown in each graph.
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2-6 Special Functions
.
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 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 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
.
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Write
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function.
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2-6 Special
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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 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.
Graph each function. Identify the domain and range.
21. SOLUTION: D = {all real numbers}
R = {all integers}
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D = {all real numbers}
R = {all integers}
2-6 Special
Functions
23. 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 range.
25. SOLUTION: D = {all real numbers
}
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D = {all real numbers
}
2-6 Special
Functions
27. SOLUTION: D = {all real numbers}
29. SOLUTION: D = {all real numbers}
31. CCSS SENSE-MAKING A 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
.
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b. Since the speed of the car cannot be negative, the appropriate domain for the function is {a | a ≥ 0}.
c.
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D = {all real numbers}
2-6 Special Functions
31. CCSS SENSE-MAKING A 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
.
b. Since the speed of the car cannot be negative, the appropriate domain for the function is {a | a ≥ 0}.
c.
Use each graph to write the absolute value function.
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
.
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Graph each function. Identify the domain and range.
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2-6 Special Functions
Use each graph to write the absolute value function.
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
.
Graph each function. Identify the domain and range.
35. SOLUTION: D = {all real numbers}
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D = {all real numbers}
2-6 Special
Functions
37. SOLUTION: D = {all real numbers}
.
41. CHALLENGE Graph
SOLUTION: 43. OPEN ENDED Write an absolute value function in which f (5) = –3.
SOLUTION: Sample answer:
45. SHORT RESPONSE What expression gives the nth term of the linear pattern defined by the table?
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SOLUTION: SampleFunctions
answer:
2-6 Special
45. SHORT RESPONSE What expression gives the nth term of the linear pattern defined by the table?
SOLUTION: So, the nth term is 3n + 1.
47. NUMBER THEORY Twelve consecutive integers 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?
F 345
G 381
H 387
J 417
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, 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.
Write an equation in slope-intercept form for the line described.
51. passes through (4, 0), parallel to 3x + 2y = 6
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the lineby3x
+ 2y =
6 is
.
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67 + 68 + 69 + 70 + 71 + 72 = 417
Therefore,
option J is the correct answer.
2-6 Special
Functions
Write an equation in slope-intercept form for the line described.
51. passes through (4, 0), parallel to 3x + 2y = 6
SOLUTION: The slope of the line 3x + 2y = 6 is
.
Therefore, the slope of a line parallel to the line 3x + 2y = 6 is
.
for m in the slope-intercept form.
Substitute
Substitute 4 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 is .
2
2
Find each value if f (x) = –4x + 6, g(x) = –x , and h(x) = –2x – 6x + 9.
53. f (2c)
SOLUTION: Substitute 2c for x in the function f (x).
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