Download 2-1 Power and Radical Functions

2-1 Power and Radical Functions
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and
where the function is increasing or decreasing.
35. SOLUTION: Evaluate the function for several x-values in its domain.
x
−300
−200
−100
0
100
200
300
g(x)
8.5
7.1
−5.9
−8
−9
−9.7
−10.2
Use these points to construct a graph.
The function is a positive odd-degree root, so there is no restriction on the domain. D = (–∞, ∞).
All values of y are included in the graph, so R = (–∞, ∞).
Solve for x when g(x) is 0 to find the x-intercept.
Substitute 0 in for x in the original equation to determine the y -intercept.
The x-intercept is −128 and the y-intercept is −8.
TheManual
y-value
approaches
positive
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positive infinity, so
infinity as x approaches negative infinity, and negative infinity as x approaches
and
.
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2-1 The
Power
and Radical
x-intercept
is −128 andFunctions
the y-intercept is −8.
The y-value approaches positive infinity as x approaches negative infinity, and negative infinity as x approaches
and
.
positive infinity, so
There are no breaks, holes, or gaps in the graph, so it is continuous for all real numbers.
As you read the graph from left to right, it is going down from negative infinity to positive infinity, so the graph is
decreasing on (−∞, ∞).
37. h(x) = 4 +
SOLUTION: Evaluate the function for several x-values in its domain.
x
h(x)
2
3
4
5
6
7
4
5.4
7
8
8.8
9.5
10.1
Use these points to construct a graph.
Since it is an even-degree radical function, the domain is restricted to nonnegative values for the radicand, 7x − 12.
Solve for x when the radicand is 0 to find the restriction on the domain.
The coefficient of the radical term is positive, so the function is increasing. Use this and the restriction on the domain
to determine the range. eSolutions Manual - Powered by Cognero
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coefficient
the radical
term is positive, so the function is increasing. Use this and the restriction on the domain
2-1 The
Power
and of
Radical
Functions
to determine the range. R = [4,
)
The graph never intersects either axis, so there are no x- or y-intercepts.
The y-value approaches positive infinity as x approaches positive infinity, so .
There are no breaks, holes, or gaps in the graph, so it is continuous over the domain .
As you read the graph from left to right, it is going up from −3 to positive infinity, so the graph is increasing on
.
39. SOLUTION: Evaluate the function for several x-values in its domain.
x
−6
−4
−2
0
2
4
6
f(x)
−78.1
−71.54
−63.81
−52.66
−61.27
−68.53
−76.35
Use these points to construct a graph.
The function is a positive odd-degree radical so there is no restriction on the domain: D = (− , ).
Use the maximum function or the trace function on a graphing calculator to approximate the maximum value of f (x)
at (0.28, −49).
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The function is a positive odd-degree radical so there is no restriction on the domain: D = (− , ).
2-1 Power and Radical Functions
Use the maximum function or the trace function on a graphing calculator to approximate the maximum value of f (x)
at (0.28, −49).
The range is restricted to values less than or equal to −49: R = (−
, −49].
The graph never crosses the x-axis, so there is no x-intercept.
Substitute 0 in for x in the original equation to determine the y -intercept.
The y -intercept is y ≈ −52.66.
The y-values approach negative infinity, as x approaches positive or negative infinity, so
and
.
There are no breaks, holes, or gaps in the graph, so it is continuous over all real numbers.
As you read the graph from left to right, it is going up from negative infinity to 0.28 and going down from 0.28 to
positive infinity, so the graph is increasing on (−∞, 0.28) and decreasing on (0.28, ∞).
41. g(x) =
− SOLUTION: Evaluate the function for several x-values in its domain.
x
1
3
6
12
15
18
22
g(x)
4.58
1.91
0.13
−2.58
−3.84
−5.14
−7.94
Use these points to construct a graph.
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12
15
18
22
−2.58
−3.84
−5.14
−7.94
2-1 Power and Radical Functions
Use these points to construct a graph.
Since g(x) includes two even-degree radicals, the domain is restricted to nonnegative values for each radicand, 22 −
x and 3x − 3. Solve for x when each radicand is greater than or equal to 0 to find the restrictions on the domain.
Thus, D = [1, 22]. Substitute these values for x to find the restrictions on the range.
x =1
x = 22
R=
Use the zero function or the trace function on a graphing calculator to approximate the x-intercept at (6.25, 0).
The graph never cross the y-axis, so there is no y-intercept.
There are no breaks, holes, or gaps in the graph, so it is continuous over the domain [1, 22].
As you read the graph from left to right, it is going down over the domain, so it is decreasing from (1, 22).
Solve each inequality.
63. SOLUTION: eSolutions Manual - Powered by Cognero
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The graph never cross the y-axis, so there is no y-intercept.
There are no breaks, holes, or gaps in the graph, so it is continuous over the domain [1, 22].
2-1 Power and Radical Functions
As you read the graph from left to right, it is going down over the domain, so it is decreasing from (1, 22).
Solve each inequality.
63. SOLUTION: Since each side of the equation was raised to a power, check for extraneous solutions.
Choose a number from the possible solution set, say x = −1.
Therefore, the solution is x ≥ −2.
65. SOLUTION: Since each side of the equation was raised to a power, check for extraneous solutions.
Choose a number from the possible solution set, say x = −7.
Since
, the function must be checked for restrictions on the domain. The radicand, 1 − 4x,
must be greater than or equal to 0. Solve 1 − 4x ≥ 0 for x.
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2-1 Therefore,
Power and
Radical
the solution
is x ≥Functions
−2.
65. SOLUTION: Since each side of the equation was raised to a power, check for extraneous solutions.
Choose a number from the possible solution set, say x = −7.
Since
, the function must be checked for restrictions on the domain. The radicand, 1 − 4x,
must be greater than or equal to 0. Solve 1 − 4x ≥ 0 for x.
The solution accounts for this restriction. Therefore, the solution is x ≤ −6.
67. SOLUTION: Since each side of the equation was raised to a power, check for extraneous solutions.
Choose a number from the possible solution set, say x = 6.
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2-1 Power and Radical Functions
The solution accounts for this restriction. Therefore, the solution is x ≤ −6.
67. SOLUTION: Since each side of the equation was raised to a power, check for extraneous solutions.
Choose a number from the possible solution set, say x = 6.
Therefore, the solution is x ≥ 5.
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