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.
1. f (x) = 5x2
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
f (x)
45
20
5
0
5
20
Use these points to construct a graph.
3
45
The function is a monomial with an even degree and a positive value for a.
D = (− , ), R = [0, ); intercept: 0;
continuous for all real numbers; decreasing: (−
, 0); increasing: (0,
)
2. g(x) = 8x5
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
−1944 −256 −8
0
8
256 1944
Use these points to construct a graph.
The function is a monomial with an odd degree and a positive value for a.
D = (− , ), R = (− , ); intercept: 0;
increasing: (−
,
continuous for all real numbers; )
3. h(x) = −x3
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
27
8
1
0
−1
−8
−27
Use these points to construct a graph.
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Page 1
The function is a monomial with an odd degree and a positive value for a.
D = (− , ), R = (− , ); intercept: 0;
2-1 Power and Radical Functions
increasing: (−
,
continuous for all real numbers; )
3. h(x) = −x3
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
27
8
1
0
−1
−8
−27
Use these points to construct a graph.
The function is a monomial with an odd degree and a negative value for a.
D = (− , ), R = (− , ); intercept: 0;
decreasing: (−
,
continuous for all real numbers; )
4. f (x) = −4x4
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
−324 −64
−4
0
−4
−64 −324
Use these points to construct a graph.
The function is a monomial with an even degree and a negative value for a.
D = (− , ), R = (− , 0]; intercept: 0;
continuous for all real numbers; increasing: (−
5. g(x) =
x
, 0); decreasing: (0,
)
9
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
−6561 −170.7 −0.3
0
0.3 170.7 6561
Use these points to construct a graph.
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Page 2
The function is a monomial with an even degree and a negative value for a.
D = (− , ), R = (− , 0]; intercept: 0;
continuous for all real numbers; 2-1 Power and Radical Functions
increasing: (−
5. g(x) =
x
, 0); decreasing: (0,
)
9
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
−6561 −170.7 −0.3
0
0.3 170.7 6561
Use these points to construct a graph.
The function is a monomial with an odd degree and a positive value for a.
D = (− , ), R = (− , ); intercept: 0;
increasing: (−
6. f (x) =
x
,
continuous for all real numbers; )
8
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
4100.6 160 0.625
0
0.625 160 4100.6
Use these points to construct a graph.
The function is a monomial with an even degree and a positive value for a.
D = (− , ), R = [0, ); intercept: 0;
continuous for all real numbers; decreasing: (−
, 0); increasing: (0,
)
7. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
1093.5 64
0.5
0
−0.5 −64 −1093.5
Use these points to construct a graph.
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Page 3
The function is a monomial with an even degree and a positive value for a.
D = (− , ), R = [0, ); intercept: 0;
continuous for all real numbers; 2-1 Power and Radical Functions
decreasing: (−
, 0); increasing: (0,
)
7. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
1093.5 64
0.5
0
−0.5 −64 −1093.5
Use these points to construct a graph.
The function is a monomial with an odd degree and a negative value for a.
D = (− , ), R = (− , ); intercept: 0;
decreasing: (−
,
continuous for all real numbers; )
8. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
−182.3 −16 −0.25
0
−0.25 −16 −182.3
Use these points to construct a graph.
The function is a monomial with an even degree and a negative value for a.
D = (− , ), R = (− , 0]; intercept: 0;
continuous for all real numbers; increasing: (−
, 0); decreasing: (0,
)
9. f (x) = 2x −4
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.025 0.125
2
2 0.125 0.025
Use these points to construct a graph.
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Page 4
The function is a monomial with an even degree and a negative value for a.
D = (− , ), R = (− , 0]; intercept: 0;
continuous for all real numbers; 2-1 Power and Radical Functions
increasing: (−
, 0); decreasing: (0,
)
9. f (x) = 2x −4
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.025 0.125
2
2 0.125 0.025
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (0, ); no intercepts;
increasing: (−
, 0); decreasing: (0,
infinite discontinuity at x = 0;
)
10. h(x) = −3x −7
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.001 0.023
3
−3 −0.023 −0.001
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (− , 0) (0, ); no intercepts;
discontinuity at x = 0; increasing: (−
, 0) and (0,
infinite )
11. f (x) = −8x −5
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.03
0.25
8
−8
−0.25 −0.03
Use these points to construct a graph.
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Page 5
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (− , 0) (0, ); no intercepts;
2-1 Power and Radical Functions
discontinuity at x = 0; increasing: (−
, 0) and (0,
infinite )
11. f (x) = −8x −5
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.03
0.25
8
−8
−0.25 −0.03
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (− , 0) (0, ); no intercepts;
discontinuity at x = 0; increasing: (−
, 0) and (0,
infinite )
12. g(x) = 7x −2
SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.78
1.75
7
7
1.75
0.78
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (0, ); no intercepts;
increasing: (−
, 0); decreasing: (0,
infinite discontinuity at x = 0;
)
13. SOLUTION: Evaluate the function for several x-values in its domain.
x
−1.5
−1
−0.5
0
0.5
1
1.5
f (x)
0.01 0.4 204.8
−204.8 −0.4 −0.01
Use these points to construct a graph.
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Page 6
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (0, ); no intercepts;
2-1 Power and Radical Functions
increasing: (−
, 0); decreasing: (0,
infinite discontinuity at x = 0;
)
13. SOLUTION: Evaluate the function for several x-values in its domain.
x
−1.5
−1
−0.5
0
0.5
1
1.5
f (x)
0.01 0.4 204.8
−204.8 −0.4 −0.01
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (−∞, 0) (0, ∞); no intercepts;
at x = 0; increasing: (−
14. h(x) =
x
, 0) and (0,
infinite discontinuity
)
−6
SOLUTION: Evaluate the function for several x-values in its domain.
x
−1.5
−1
−0.5 0
0.5
1
1.5
f (x)
0.01
0.17 10.67 10.67 0.17
0.01
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (0, ); no intercepts;
increasing: (−
15. h(x) =
x
, 0); decreasing: (0,
infinite discontinuity at x = 0;
)
−3
SOLUTION: Evaluate the function for several x-values in its domain.
x
−1.5
−1
−0.5
0
0.5
1
1.5
f (x)
−0.22 −0.75 −6
6
0.75
0.22
Use these points to construct a graph.
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Page 7
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (0, ); no intercepts;
2-1 Power and Radical Functions
increasing: (−
15. h(x) =
x
, 0); decreasing: (0,
infinite discontinuity at x = 0;
)
−3
SOLUTION: Evaluate the function for several x-values in its domain.
x
−1.5
−1
−0.5
0
0.5
1
1.5
f (x)
−0.22 −0.75 −6
6
0.75
0.22
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (− , 0) (0, ); no intercepts;
discontinuity at x = 0; decreasing: (−
, 0) and (0,
infinite )
16. SOLUTION: Evaluate the function for several x-values in its domain.
x
−1.5
−1
−0.5
0
0.5
1
1.5
f (x)
−0.03 −0.7 −179.2 −179.2 −0.7 −0.03
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (− , 0); no intercepts;
decreasing: (−
, 0); increasing: (0,
infinite discontinuity at x = 0;
)
17. GEOMETRY The volume of a sphere is given by V(r) =
3
πr , where r is the radius.
a. State the domain and range of the function.
b. Graph the function.
SOLUTION: a. The radius of a sphere cannot have a negative length. The radius also cannot be 0 because then the object would
fail to be a sphere. Thus, D = (0, ), R = (0, )
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Page 8
b. Evaluate the function for several x-values in its domain.
x
0.5
1
1.5
2
2.5
3
3.5
f (x)
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (− , 0); no intercepts;
2-1 Power and Radical Functions
decreasing: (−
, 0); increasing: (0,
infinite discontinuity at x = 0;
)
17. GEOMETRY The volume of a sphere is given by V(r) =
3
πr , where r is the radius.
a. State the domain and range of the function.
b. Graph the function.
SOLUTION: a. The radius of a sphere cannot have a negative length. The radius also cannot be 0 because then the object would
fail to be a sphere. Thus, D = (0, ), R = (0, )
b. Evaluate the function for several x-values in its domain.
x
0.5
1
1.5
2
2.5
3
3.5
f (x)
0.5
4.2 14.1 33.5
65.5 113.1 179.6
Use these points to construct a graph.
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and
where the function is increasing or decreasing.
18. SOLUTION: Evaluate the function for several x-values in its domain.
x
0
1
2
3
4
5
6
f (x)
0
8
9.5 10.5
11.3
12
12.5
Use these points to construct a graph.
Since the denominator of the power is even, the domain must be restricted to nonnegative values.
D = [0, ), R = [0, ); intercept: 0;
; continuous on [0, ); increasing: (0, )
19. SOLUTION: Evaluate the function for several x-values in its domain.
x
−6
−4
−2
0
2
4
6
f (x)
8.6 by
7.9Cognero6.9
0
−6.9
−7.9 −8.6
eSolutions
Manual - Powered
Use these points to construct a graph.
Page 9
2-1
Since the denominator of the power is even, the domain must be restricted to nonnegative values.
D = [0, and
), R Radical
= [0, ); intercept:
0;
; continuous on [0, ); increasing: (0, )
Power
Functions
19. SOLUTION: Evaluate the function for several x-values in its domain.
x
−6
−4
−2
0
2
4
6
f (x)
8.6
7.9
6.9
0
−6.9
−7.9 −8.6
Use these points to construct a graph.
D = (−
,
), R = (−
decreasing: (−
,
,
continuous for all real numbers; ); intercept: 0;
)
20. SOLUTION: Evaluate the function for several x-values in its domain.
x
−1.5
−1
−0.5
0
0.5
1
1.5
f (x)
0.17 0.2
0.25
−0.25 −0.2 −0.17
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (− , 0) (0, ); no intercepts;
discontinuity at x = 0; increasing: (−
, 0) and (0,
infinite )
21. SOLUTION: Evaluate the function for several x-values in its domain.
x
1
2
3
4
5
6
7
f (x)
10.0 8.9
8.3
7.9
7.6
7.4
7.2
Use these points to construct a graph.
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Page 10
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (− , 0) (0, ); no intercepts;
2-1 Power and Radical Functions
discontinuity at x = 0; increasing: (−
, 0) and (0,
infinite )
21. SOLUTION: Evaluate the function for several x-values in its domain.
x
1
2
3
4
5
6
7
f (x)
10.0 8.9
8.3
7.9
7.6
7.4
7.2
Use these points to construct a graph.
Since the denominator of the power is even and the power is negative, the domain must be restricted to positive
values.
D = (0, ), R = (0, ); no intercepts;
; continuous on (0, ); decreasing: (0, )
22. SOLUTION: Evaluate the function for several x-values in its domain.
x
0
1
2
3
4
5
6
f (x)
0
−3
−4.6 −6
−7.1
−8.2 −9.2
Use these points to construct a graph.
Since the denominator of the power is even, the domain must be restricted to nonnegative values.
D = [0, ), R = [0, ); intercept: 0;
; continuous on [0, ); decreasing: (0, )
23. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
−1.45 −1.13 −0.75
0
0.75
1.13 1.45
Use these points to construct a graph.
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Page 11
Since the denominator of the power is even, the domain must be restricted to nonnegative values.
; continuous on [0, ); decreasing: (0, )
2-1 Power
Functions
D = [0, and
), R Radical
= [0, ); intercept:
0;
23. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
−1.45 −1.13 −0.75
0
0.75
1.13 1.45
Use these points to construct a graph.
D = (−
,
), R = (−
increasing: (−
,
,
); intercept: 0;
continuous for all real numbers; )
24. SOLUTION: Evaluate the function for several x-values in its domain.
x
0.5
1
1.5
2
2.5
3
3.5
f (x)
−0.84 −0.5 −0.37 −0.3 −0.25 −0.22 −0.2
Use these points to construct a graph.
Since the denominator of the power is even and the power is negative, the domain must be restricted to positive
values.
D = (0, ), R = (− , 0); no intercept;
; continuous on (0, ); increasing: (0, )
25. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.48 0.63
1
1
0.63 0.48
Use these points to construct a graph.
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Page 12
2-1
Since the denominator of the power is even and the power is negative, the domain must be restricted to positive
values.
Power
D = (0, and
), R Radical
= (− , 0); Functions
no intercept;
; continuous on (0, ); increasing: (0, )
25. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.48 0.63
1
1
0.63 0.48
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (0, ); no intercepts;
increasing: (−
, 0); decreasing: (0,
infinite discontinuity at x = 0;
)
26. SOLUTION: Evaluate the function for several x-values in its domain.
x
−1.5
−1
−0.5
0
0.5
1
1.5
f (x)
−13.8 −7
−2.2
0
2.2
7
13.8
Use these points to construct a graph.
D = (−
,
), R = (−
increasing: (−
,
,
); intercept: 0;
continuous for all real numbers; )
27. SOLUTION: Evaluate the function for several x-values in its domain.
x
0
0.5
1
1.5
2
2.5
3
f (x)
0
−1.2
−4 −8.1 −13.5 −19.9 −27.4
Use these points to construct a graph.
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Page 13
D = (−
,
), R = (−
,
); intercept: 0;
2-1 Power and Radical Functions
increasing: (−
,
continuous for all real numbers; )
27. SOLUTION: Evaluate the function for several x-values in its domain.
x
0
0.5
1
1.5
2
2.5
3
f (x)
0
−1.2
−4 −8.1 −13.5 −19.9 −27.4
Use these points to construct a graph.
Since the denominator of the power is even, the domain must be restricted to nonnegative values.
D = [0, ), R = (− , 0]; intercept: 0;
; continuous on [0, ); decreasing: (0, )
28. SOLUTION: Evaluate the function for several x-values in its domain.
x
0.5
1
1.5
2
2.5
3
3.5
f (x)
−14.1 −5
−2.7 −1.8 −1.3
−1
−0.8
Use these points to construct a graph.
Since the denominator of the power is even and the power is negative, the domain must be restricted to positive
values.
D = (0, ), R = (− , 0); no intercepts;
; continuous on (0, ); increasing: (0, )
29. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.11 0.22
0.67
0.67
0.22 0.11
Use these points to construct a graph.
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Page 14
2-1
Since the denominator of the power is even and the power is negative, the domain must be restricted to positive
values.
Power
D = (0, and
), R Radical
= (− , 0); Functions
no intercepts;
; continuous on (0, ); increasing: (0, )
29. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
0.11 0.22
0.67
0.67
0.22 0.11
Use these points to construct a graph.
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (0, ); no intercepts;
increasing: (−
infinite discontinuity at x = 0;
, 0); decreasing: (0, ∞)
Complete each step.
a. Create a scatter plot of the data.
b. Determine a power function to model the data.
c. Calculate the value of each model at x = 30.
30. SOLUTION: a. Enter the data into a graphing calculator and create a scatter plot.
b. Use the power regression function on the graphing calculator to find values for a and b.
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2.89
Page 15
Since the power is negative, the function will be undefined at x = 0.
D = (− , 0) (0, ), R = (0, ); no intercepts;
2-1 Power and Radical Functions
increasing: (−
infinite discontinuity at x = 0;
, 0); decreasing: (0, ∞)
Complete each step.
a. Create a scatter plot of the data.
b. Determine a power function to model the data.
c. Calculate the value of each model at x = 30.
30. SOLUTION: a. Enter the data into a graphing calculator and create a scatter plot.
b. Use the power regression function on the graphing calculator to find values for a and b.
2.89
y = 3.54x
c. Graph the regression equation using a graphing calculator. To calculate x = 30, use the CALC function on the
graphing calculator.
The value of the model at x = 30 is about 66,098.82.
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31. Page 16
2-1 Power
and
Radical
The value
of the
model at Functions
x = 30 is about 66,098.82.
31. SOLUTION: a. Enter the data into a graphing calculator and create a scatter plot.
b. Use the power regression function on the graphing calculator to find values for a and b.
5.75
y = 0.77x .
c. Graph the regression equation using a graphing calculator. To calculate x = 30, use the CALC function on the
graphing calculator.
The value of the model at x = 30 is about 235,906,039.
32. CLIFF DIVING In the sport of cliff diving, competitors perform three dives from a height of 28 meters. Judges award divers a score from 0 to 10 points based on degree of difficulty, take-off, positions, and water entrance. The
table shows the speed of a diver at various distances in the dive.
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Page 17
2-1 Power
and Radical Functions
The value of the model at x = 30 is about 235,906,039.
32. CLIFF DIVING In the sport of cliff diving, competitors perform three dives from a height of 28 meters. Judges award divers a score from 0 to 10 points based on degree of difficulty, take-off, positions, and water entrance. The
table shows the speed of a diver at various distances in the dive.
a. Create a scatter plot of the data.
b. Determine a power function to model the data.
c. Use the function to predict the speed at which a diver would enter the water from a cliff dive of 30 meters.
SOLUTION: a. Enter the data into a graphing calculator and create a scatter plot.
b. Use the power regression function on the graphing calculator to find values for a and b.
0.5
f(x) = 4.42x .
c. Graph the regression equation using a graphing calculator. To calculate the speed at which a diver would enter
the water from a cliff dive of 30 meters, use the CALC function on the graphing calculator. Let x = 30.
The speed at which a diver would enter the water from a cliff dive of 30 meters is about 24.25 meters per second.
33. WEATHER The wind chill temperature is the apparent temperature felt on exposed skin, taking into account the effect of the wind. The table shows the wind chill temperature produced at winds of various speeds when the
actual temperature is 50ºF.
eSolutions Manual - Powered by Cognero
Page 18
2-1 Power
and Radical Functions
The speed at which a diver would enter the water from a cliff dive of 30 meters is about 24.25 meters per second.
33. WEATHER The wind chill temperature is the apparent temperature felt on exposed skin, taking into account the effect of the wind. The table shows the wind chill temperature produced at winds of various speeds when the
actual temperature is 50ºF.
a. Create a scatter plot of the data.
b. Determine a power function to model the data.
c. Use the function to predict the wind chill temperature when the wind speed is 65 miles per hour.
SOLUTION: a. Enter the data into a graphing calculator and create a scatter plot.
b. Use the power regression function on the graphing calculator to find values for a and b.
−0.0797
f(x) = 55.14x
.
c. Graph the regression equation using a graphing calculator. To predict the wind chill temperature when the wind
speed is 65 miles per hour, use the CALC function on the graphing calculator. Let x = 65.
The wind chill temperature when the wind speed is 65 miles per hour is about 39.54°F
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and
where the function is increasing or decreasing.
34. f (x) = 3
SOLUTION: eSolutions Manual - Powered by Cognero
Evaluate the function for several x-values in its domain.
x
−2
−1
0
1
2
3
Page 19
4
2-1 Power
and Radical Functions
The wind chill temperature when the wind speed is 65 miles per hour is about 39.54°F
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and
where the function is increasing or decreasing.
34. f (x) = 3
SOLUTION: Evaluate the function for several x-values in its domain.
x
−2
−1
0
1
2
3
4
f (x)
0
5.2
7.3
9
10.4
11.6 12.7
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, 6 + 3x.
Solve for x when the radicand is 0 to find the restriction on the domain and the x-intercept.
), R = [0, ∞); x-intercept: −2, y-intercept: 3
D = [−2,
;
continuous on [−2,
); increasing: (−2,
)
35. SOLUTION: Evaluate the function for several x-values in its domain.
x
−300 −200 −100
0
100
200
300
f (x)
8.5
7.1 −5.9 −8
−9
−9.7 −10.2
Use these points to construct a graph.
Solve for x when g(x) is 0 to find the x-intercept.
eSolutions Manual - Powered by Cognero
D = (−
,
), R = (−
,
Page 20
); x-intercept: −128, y-intercept: −8;
all real numbers; decreasing: (−
,
)
continuous for ), R = [0, ∞); x-intercept: −2, y-intercept: 3
D = [−2,
;
continuous on [−2,
); increasing: (−2,
2-1 Power
and Radical Functions
)
35. SOLUTION: Evaluate the function for several x-values in its domain.
x
−300 −200 −100
0
100
200
300
f (x)
8.5
7.1 −5.9 −8
−9
−9.7 −10.2
Use these points to construct a graph.
Solve for x when g(x) is 0 to find the x-intercept.
D = (−
,
), R = (−
,
); x-intercept: −128, y-intercept: −8;
all real numbers; decreasing: (−
,
continuous for )
36. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
−3 −3.6 −3.67 −3.71 −3.75 −3.78 −3.8
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, 16x +
48. Solve for x when the radicand is 0 to find the x-intercept to find the restriction on the domain.
D = [−3,
), R = (−
, −3]; x-intercept: none, y-intercept: −3.71;
eSolutions Manual - Powered by Cognero
decreasing: (−3,
37. h(x) = 4 +
)
continuous on [−3,
);
Page 21
D = (−
,
), R = (−
,
); x-intercept: −128, y-intercept: −8;
continuous for 2-1 Power and Radical Functions
all real numbers; decreasing: (−
,
)
36. SOLUTION: Evaluate the function for several x-values in its domain.
x
−3
−2
−1
0
1
2
3
f (x)
−3 −3.6 −3.67 −3.71 −3.75 −3.78 −3.8
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, 16x +
48. Solve for x when the radicand is 0 to find the x-intercept to find the restriction on the domain.
D = [−3,
), R = (−
decreasing: (−3,
, −3]; x-intercept: none, y-intercept: −3.71;
continuous on [−3,
);
)
37. h(x) = 4 +
SOLUTION: Evaluate the function for several x-values in its domain.
x
2
3
4
5
6
f (x)
4
5.4
7
8
Use these points to construct a graph.
8.8
9.5
7
10.1
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.
, R =by[4,Cognero
); no
eSolutions Manual - Powered
38. intercepts;
continuous on ; increasing:
Page 22
D = [−3,
), R = (−
, −3]; x-intercept: none, y-intercept: −3.71;
2-1 Power and Radical Functions
decreasing: (−3,
continuous on [−3,
);
)
37. h(x) = 4 +
SOLUTION: Evaluate the function for several x-values in its domain.
x
2
3
4
5
6
f (x)
4
5.4
7
8
Use these points to construct a graph.
8.8
9.5
7
10.1
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.
, R = [4,
38. g(x) =
); no intercepts;
continuous on ; increasing:
– 16
SOLUTION: Evaluate the function for several x-values in its domain.
x
−2.5
−2
−1.5 −1
−0.5
0
0.25
f (x)
20.5
11
2.5 −4.8 −10.8 −15
−16
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, (1 − 4x)
3
. Solve for x when the radicand is 0 to find the restriction on the domain.
eSolutions Manual - Powered by Cognero
Solve for x when g(x) is 0 to find the x-intercept.
Page 23
2-1 Power and, RRadical
= [4, );Functions
no intercepts;
38. g(x) =
continuous on ; increasing:
– 16
SOLUTION: Evaluate the function for several x-values in its domain.
x
−2.5
−2
−1.5 −1
−0.5
0
0.25
f (x)
20.5
11
2.5 −4.8 −10.8 −15
−16
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, (1 − 4x)
3
. Solve for x when the radicand is 0 to find the restriction on the domain.
Solve for x when g(x) is 0 to find the x-intercept.
D = (−
, 0.25], R = [−16,
decreasing: (−
); x-intercept: −1.34, y-intercept: −15;
continuous on (−
, 0.25];
, 0.25)
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 −69.53 −76.35
Use these points to construct a graph.
eSolutions Manual - Powered by Cognero
Page 24
D = (−
, 0.25], R = [−16,
); x-intercept: −1.34, y-intercept: −15;
2-1 Power and Radical Functions
decreasing: (−
continuous on (−
, 0.25];
, 0.25)
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 −69.53 −76.35
Use these points to construct a graph.
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. Also, there is a turning point at x = 0.28.
D = (− , ), R = (− , −49.00]; x-intercept: none, y-intercept: −52.66;
continuous for all real numbers; increasing: (− , 0.28); decreasing: (0.28,
)
40. SOLUTION: Evaluate the function for several x-values in its domain.
x
−8
−4
0
4
8
12
16
f (x)
−6.2 −6.4 −6.5 −6.7 −6.9
−7.3 −8.9
Use these points to construct a graph.
Solve for x when h(x) is 0 to find the x-intercept.
eSolutions Manual - Powered by Cognero
Page 25
2-1
The range is restricted to values less than or equal to −49. Also, there is a turning point at x = 0.28.
D = (− , ), R = (− , −49.00]; x-intercept: none, y-intercept: −52.66;
Power and Radical Functions
continuous for all real numbers; increasing: (− , 0.28); decreasing: (0.28,
)
40. SOLUTION: Evaluate the function for several x-values in its domain.
x
−8
−4
0
4
8
12
16
f (x)
−6.2 −6.4 −6.5 −6.7 −6.9
−7.3 −8.9
Use these points to construct a graph.
Solve for x when h(x) is 0 to find the x-intercept.
D = (−
,
), R = (−
,
); x-intercept: −2034.5, y-intercept: −6.5;
for all real numbers; decreasing: (−
41. g(x) =
,
continuous
)
− SOLUTION: Evaluate the function for several x-values in its domain.
x
1
3
6
12
15
18
22
f (x)
4.58 1.91 0.13 −2.58 −3.84 −5.14 −7.94
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.
eSolutions Manual - Powered by Cognero
Thus, x must be 1 ≤ x ≤ 22. Substitute these values for x to find the restrictions on the range.
Page 26
D = (−
,
), R = (−
,
); x-intercept: −2034.5, y-intercept: −6.5;
2-1 Power and Radical Functions
for all real numbers; decreasing: (−
41. g(x) =
,
continuous
)
− SOLUTION: Evaluate the function for several x-values in its domain.
x
1
3
6
12
15
18
22
f (x)
4.58 1.91 0.13 −2.58 −3.84 −5.14 −7.94
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, x must be 1 ≤ x ≤ 22. Substitute these values for x to find the restrictions on the range.
x =1
x = 22
Use the zero function or the trace function on a graphing calculator to approximate the x-intercept at (6.25, 0).
D = [1, 22], R = [–
,
]; x-intercept: 6.25; continuous on [1, 22]; decreasing: (1, 22)
42. FLUID MECHANICS The velocity of the water flowing through a hose with a nozzle can be modeled using V
(P) = 12.1
, where V is the velocity in feet per second and P is the pressure in pounds per square inch.
a. Graph the velocity through a nozzle as a function of pressure.
b. Describe the domain, range, end behavior, and continuity of the function and determine where it is increasing or
decreasing.
SOLUTION: eSolutions
Manual -the
Powered
by Cognero
Evaluate
function
for several
x
f (x)
0
0
1
12.1
2
17.1
x-values in its domain.
3
4
5
6
21
24.2
27.1 29.7
Page 27
2-1 Power and Radical Functions
D = [1, 22], R = [–
,
]; x-intercept: 6.25; continuous on [1, 22]; decreasing: (1, 22)
42. FLUID MECHANICS The velocity of the water flowing through a hose with a nozzle can be modeled using V
(P) = 12.1
, where V is the velocity in feet per second and P is the pressure in pounds per square inch.
a. Graph the velocity through a nozzle as a function of pressure.
b. Describe the domain, range, end behavior, and continuity of the function and determine where it is increasing or
decreasing.
SOLUTION: Evaluate the function for several x-values in its domain.
x
0
1
2
3
4
5
6
f (x)
0
12.1 17.1
21
24.2
27.1 29.7
Use these points to construct a graph.
b. Since it is an even-degree radical function, the domain is restricted to nonnegative values for the radicand, P.
Thus, P ≥ 0.
D = [0, ), R = [0, );
; continuous on [0, ); increasing: (0, )
43. AGRICULTURAL SCIENCE The net energy NEm required to maintain the body weight of beef cattle, in
megacalories (Mcal) per day, is estimated by the formula
where m is the animal’s mass in
kilograms. One megacalorie is equal to one million calories.
a. Find the net energy per day required to maintain a 400-kilogram steer.
b. If 0.96 megacalorie of energy is provided per pound of whole grain corn, how much corn does a 400-kilogram
steer need to consume daily to maintain its body weight?
SOLUTION: a. Substitute m = 400.
The net energy per day required to maintain a 400-kilogram steer is approximately 6.89 Mcal.
b. Divide 6.89 Mcal by the 0.96 Mcal found in a pound of whole grain corn to find the total amount of corn
necessary to maintain a 400-kilogram steer.
It will take about 7.18 pounds of corn to maintain a 400-kilogram steer.
Solve each equation.
44. 4 =
+ eSolutions Manual - Powered by Cognero
SOLUTION: Page 28
necessary to maintain a 400-kilogram steer.
2-1 Power and Radical Functions
It will take about 7.18 pounds of corn to maintain a 400-kilogram steer.
Solve each equation.
44. 4 =
+ SOLUTION: Since the each side of the equation was raised to a power, check the solutions in the original equation.
x = − 11
x= −75
Neither value for x is a solution for the original equation. Thus, there is no solution.
45. 0.5x =
+ 2
SOLUTION: Since the each side of the equation was raised to a power, check the solutions in the original equation.
x =0
eSolutions
x=Manual
−4 - Powered by Cognero
Page 29
2-1 Power
and for
Radical
Functions
Neither value
x is a solution
for the original equation. Thus, there is no solution.
45. 0.5x =
+ 2
SOLUTION: Since the each side of the equation was raised to a power, check the solutions in the original equation.
x =0
x= −4
Neither value for x is a solution for the original equation. Thus, there is no solution.
46. −3 =
− SOLUTION: Since the each side of the equation was raised to a power, check the solutions in the original equation.
x = 13
eSolutions Manual - Powered by Cognero
x= 1.75
Page 30
2-1 Power and Radical Functions
Neither value for x is a solution for the original equation. Thus, there is no solution.
46. −3 =
− SOLUTION: Since the each side of the equation was raised to a power, check the solutions in the original equation.
x = 13
x= 1.75
One solution checks and the other solution does not. Therefore, the solution is x = 13.
47. − 10 = 17
SOLUTION: Since the each side of the equation was raised to a power, check the solution in the original equation.
The solution is x = 7.
48. eSolutions Manual - Powered by Cognero
SOLUTION: Page 31
2-1 Power and Radical Functions
The solution is x = 7.
48. SOLUTION: Since the each side of the equation was raised to a power, check the solution in the original equation.
The solution is x = 23.
49. x =
+ 2
SOLUTION: Since the each side of the equation was raised to a power, check the solutions in the original equation.
x =2
x= 4
The solutions are x = 2 and x = 4.
50. 7 +
= 250
SOLUTION: eSolutions Manual - Powered by Cognero
Page 32
2-1 Power and Radical Functions
The solutions are x = 2 and x = 4.
50. 7 +
= 250
SOLUTION: Since the each side of the equation was raised to a power, check the solution in the original equation.
The solution is x = −9.
51. x = 5 +
SOLUTION: Since the each side of the equation was raised to a power, check the solutions in the original equation.
x =8
x= 3
One solution checks and the other solution does not. Therefore, the solution is x = 8.
52. +4=
SOLUTION: eSolutions Manual - Powered by Cognero
Page 33
2-1 Power and Radical Functions
One solution checks and the other solution does not. Therefore, the solution is x = 8.
52. +4=
SOLUTION: Since the each side of the equation was raised to a power, check the solutions in the original equation.
x =2
x= 10
The solutions are x = 2 and x = 10.
53. = − 20
SOLUTION: Since the each side of the equation was raised to a power, check the solution in the original equation.
There is no solution.
54. − 1 =
SOLUTION: eSolutions Manual - Powered by Cognero
Page 34
2-1 Power and Radical Functions
There is no solution.
54. − 1 =
SOLUTION: Since the each side of the equation was raised to a power, check the solutions in the original equation.
x = −1
One solution checks and the other solution does not. Therefore, the solution is x = −1.
55. SOLUTION: Since the each side of the equation was raised to a power, check the solution in the original equation.
The solution is x = 10.
Determine whether each function is a monomial function given that a and b are positive integers.
Explain your reasoning.
eSolutions Manual4a- Powered by Cognero
56. y =
x
Page 35
2-1 Power and Radical Functions
The solution is x = 10.
Determine whether each function is a monomial function given that a and b are positive integers.
Explain your reasoning.
56. y =
x
4a
SOLUTION: n
Yes; sample answer: The function follows the form f (x) = ax , where n is a positive integer. In this case, a =
and n = 4a.
57. G(x) = –2ax4
SOLUTION: n
Yes; sample answer: The function follows the form f (x) = ax , where n is a positive integer. In this case, a = –2a
and n = 4.
58. F(b) = 3ab5x
SOLUTION: No; sample answer: The function is not a power function because the variable is in the exponent.
59. y =
ab
t
SOLUTION: n
Yes; sample answer: The function follows the form f (x) = ax , where n is a positive integer. In this case, a =
and n = ab.
60. SOLUTION: n
Yes; sample answer: The function follows the form f (x) = ax , where n is a positive integer. In this case, a =
and n = 2b.
61. y = 4abx–2
SOLUTION: No; sample answer: The function is not a monomial function because the exponent for x is negative.
62. CHEMISTRY The function can be used to approximate the nuclear radius of an element based on −15
its molecular mass, where r is length of the radius in meters, R0 is a constant (about 1.2 × 10
meter), and A is
the molecular mass.
eSolutions Manual - Powered by Cognero
Page 36
61. y = 4abx–2
SOLUTION: 2-1 Power
and Radical Functions
No; sample answer: The function is not a monomial function because the exponent for x is negative.
62. CHEMISTRY The function can be used to approximate the nuclear radius of an element based on −15
its molecular mass, where r is length of the radius in meters, R0 is a constant (about 1.2 × 10
meter), and A is
the molecular mass.
a. If the nuclear radius of sodium is about 3.412 × 10−15 meter, what is its molecular mass?
−15
b. The approximate nuclear radius of an element is 6.030 × 10
meter. Identify the element.
c. The ratio of the molecular masses of two elements is 27:8. What is the ratio of their nuclear radii?
SOLUTION: a. Substitute r = 3.412 × 10−15 and R0 = 1.2 × 10−15 into the function to solve for A.
b. Substitute r = 6.030 × 10−15 and R0 = 1.2 × 10−15 into the function to solve for A.
With a molecular mass of 126.88, the element is iodine.
c. Although the molecular masses of the two elements are unknown, the ratio of the two masses will always be
27:8. These numbers can be used for the molecular masses to find the ratio of the radii.
Element 1
Element 2
Since R0 is a constant, the ratio of the radii will be 3:2.
Solve each inequality.
63. SOLUTION: eSolutions Manual - Powered by Cognero
Page 37
2-1 Power
and Radical Functions
Since R is a constant, the ratio of the radii will be 3:2.
0
Solve each inequality.
63. SOLUTION: Since the each side of the equation was raised to a power, check a solution in the original equation.
x = −1
The solution is x ≥ −2.
64. SOLUTION: Since the each side of the equation was raised to a power, check a solution in the original equation.
x =5
The solution is x ≤ 6.
65. SOLUTION: Since the each side of the equation was raised to a power, check a solution in the original equation.
x = −7
eSolutions Manual - Powered by Cognero
Page 38
2-1 Power and Radical Functions
The solution is x ≤ 6.
65. SOLUTION: Since the each side of the equation was raised to a power, check a solution in the original equation.
x = −7
Since the denominator of the exponent is even, 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. So, the solution is x ≤ −6.
66. ≤ 9
SOLUTION: Since the each side of the equation was raised to a power, check a solution in the original equation.
x = 24
Since the denominator of the exponent is even, the function must be checked for restrictions on the domain. The
radicand, 6 + 3x, must be greater than or equal to 0. Solve 6 + 3x ≥ 0 for x.
Since the solution does not account for this restriction, it must be added to the solution. The solution is −2 ≤ x ≤ 25.
eSolutions Manual - Powered by Cognero
67. Page 39
2-1 Power and Radical Functions
The solution accounts for this restriction. So, the solution is x ≤ −6.
66. ≤ 9
SOLUTION: Since the each side of the equation was raised to a power, check a solution in the original equation.
x = 24
Since the denominator of the exponent is even, the function must be checked for restrictions on the domain. The
radicand, 6 + 3x, must be greater than or equal to 0. Solve 6 + 3x ≥ 0 for x.
Since the solution does not account for this restriction, it must be added to the solution. The solution is −2 ≤ x ≤ 25.
67. SOLUTION: Since the each side of the equation was raised to a power, check a solution in the original equation.
x =6
The solution is x ≥ 5.
68. SOLUTION: eSolutions Manual - Powered by Cognero
Page 40
2-1 Power and Radical Functions
The solution is x ≥ 5.
68. SOLUTION: Since the each side of the equation was raised to a power, check a solution in the original equation.
x = 291
69. CHEMISTRY Boyle’s Law states that, at constant temperature, the pressure of a gas is inversely proportional to
its volume. The results of an experiment to explore Boyle’s Law are shown.
a. Create a scatter plot of the data.
b. Determine a power function to model the pressure P as a function of volume v.
c. Based on the information provided in the problem statement, does the function you determined in part b make
sense? Explain.
d. Use the model to predict the pressure of the gas if the volume is 3.25 liters.
e . Use the model to predict the pressure of the gas if the volume is 6 liters.
SOLUTION: a.
eSolutions Manual - Powered by Cognero
b. Use the power regression function on the graphing calculator to find values for a and n.
Page 41
sense? Explain.
d. Use the model to predict the pressure of the gas if the volume is 3.25 liters.
e . Use the model to predict the pressure of the gas if the volume is 6 liters.
2-1 Power
and Radical Functions
SOLUTION: a.
b. Use the power regression function on the graphing calculator to find values for a and n.
–1
P(v) = 3.62v
c. Sample answer: Yes; the problem states that the volume and pressure are inversely proportional, and in the
power function, the exponent of the volume variable is −1.
d. Graph the regression equation using a graphing calculator. To predict the pressure of the gas if the volume is
3.25 liters, use the CALC function on the graphing calculator. Let v = 3.25.
The pressure of 3.25 liters of the gas is about 1.12 atmospheres.
e . Graph the regression equation using a graphing calculator. To predict the pressure of the gas if the volume is 6
liters, use the CALC function on the graphing calculator. Let v = 6.
The pressure of 6 liters of the gas is about 0.60 atmospheres.
Without using a calculator, match each graph with the appropriate function.
a.
b. g(x) =
x
6
c. h(x) = 4x −3
d.
eSolutions Manual - Powered by Cognero
Page 42
2-1 Power and Radical Functions
The pressure of 6 liters of the gas is about 0.60 atmospheres.
Without using a calculator, match each graph with the appropriate function.
a.
b. g(x) =
x
6
c. h(x) = 4x −3
d.
70. SOLUTION: The end behavior of the graph indicates that n is positive and even. Also, a is positive. The equation that matches
this description is g(x) =
6
x .
The answer is b.
71. SOLUTION: There is an infinite discontinuity at x = 0. This indicates that the power is negative. The equation that matches this
−3
description is h(x) = 4x .
The answer is c.
72. SOLUTION: The domain of the graph is restricted to nonnegative values. This indicates that the function has a rational exponent
with an even denominator or is a radical function with an even value for n. The equation that matches this
description is
.
The answer is a.
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Page 43
The domain of the graph is restricted to nonnegative values. This indicates that the function has a rational exponent
with an even denominator or is a radical function with an even value for n. The equation that matches this
description is
.
2-1 Power and Radical Functions
The answer is a.
73. SOLUTION: The end behavior and the continuity of the graph indicate that it is a radical function with an odd value for n. The
equation that matches this description is
.
The answer is d.
74. ELECTRICITY The voltage used by an electrical device such as a DVD player can be calculated using V =
, where V is the voltage in volts, P is the power in watts, and R is the resistance in ohms. The function I =
can be used to calculate the current, where I is the current in amps.
a. If a lamp uses 120 volts and has a resistance of 11 ohms, what is the power consumption of the lamp?
b. If a DVD player has a current of 10 amps and consumes 1200 watts of power, what is the resistance of the
DVD player?
c. Ohm’s Law expresses voltage in terms of current and resistance. Use the equations given above to write Ohm’s
Law using voltage, resistance, and amperage.
SOLUTION: a. Substitute V = 120 and R = 11 into V =
to solve for P.
The power consumption of the lamp is 1309 watts.
b. Substitute I = 10 and P = 1200 into I =
.
The resistance of the DVD player is 12 ohms.
c. To write Ohm’s Law in terms of V, R, and I, solve I =
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for P.
Page 44
2-1 Power and Radical Functions
The resistance of the DVD player is 12 ohms.
c. To write Ohm’s Law in terms of V, R, and I, solve I =
2
Substitute P = I R into V =
for P.
.
Use the points provided to determine the power function represented by the graph.
75. SOLUTION: The function is undefined at x = 0, therefore; it is a power function of the form
where
a is a real constant and n is a natural number. Start by assuming n = 1 and solve for a by substituting a set of
points (2, 2) for x and y.
When n =1, f (x) = 4 · or for the point (2, 2). If this power function is true for the second point (−1, −4), then
f(x) can represent the graph.
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Since f (x) is true for (−1, −4), f (x) =
or 4x
−1
is a power function for the graph.
Page 45
2-1 Power and Radical Functions
Use the points provided to determine the power function represented by the graph.
75. SOLUTION: where
The function is undefined at x = 0, therefore; it is a power function of the form
a is a real constant and n is a natural number. Start by assuming n = 1 and solve for a by substituting a set of
points (2, 2) for x and y.
When n =1, f (x) = 4 · or for the point (2, 2). If this power function is true for the second point (−1, −4), then
f(x) can represent the graph.
Since f (x) is true for (−1, −4), f (x) =
or 4x
−1
is a power function for the graph.
76. SOLUTION: n
The end behavior of the function indicates that it is power function of the form f (x) = ax , where n is even and a is
positive. Start by assuming n = 2 and solve for a by substituting a set of points
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for x and y.
Page 46
SOLUTION: n
The end behavior of the function indicates that it is power function of the form f (x) = ax , where n is even and a is
2-1 Power and Radical Functions
positive. Start by assuming n = 2 and solve for a by substituting a set of points
When n =2, f (x) =
2
x for the point
for x and y.
. If this power function is true for the second point
, then f
(x) can represent the graph.
Since f (x) is not true for the second point, repeat the first step but assume that n = 4.
When n =4, f (x) =
4
x for the point
. If this power function is true for the second point
, then f
(x) can represent the graph.
Since f (x) is true for
, f (x) =
4
x is a power function for the graph.
77. SOLUTION: The restricted domain of the graph indicates that it is a power function of the form
where a is a real
number and n is an even integer. Start by assuming n = 2 and solve for a by substituting a set of points (1, 3) for x
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Page 47
and y.
2-1 Power
and
Radical
Functions
Since f (x)
is true
for
, f (x) =
4
x is a power function for the graph.
77. SOLUTION: The restricted domain of the graph indicates that it is a power function of the form
where a is a real
number and n is an even integer. Start by assuming n = 2 and solve for a by substituting a set of points (1, 3) for x
and y.
When n =2,
represent the graph.
for the point (1, 3). If this power function is true for the second point (4, 6), then f (x) can
Since f (x) is true for (4, 6),
is a power function for the graph.
78. SOLUTION: The function is undefined at x = 0, therefore, it is a power function of the form
where a is a real constant and n is a natural number. Start by assuming n = 1 and solve for a by substituting a set of
points
for x and y.
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Page 48
The function is undefined at x = 0, therefore, it is a power function of the form
where a is a real constant and n is a natural number. Start by assuming n = 1 and solve for a by substituting a set of
2-1 Power and Radical Functions
for x and y.
points
When n =1, f (x) =
· or for the point . If this power function is true for the second point
, then f (x) can represent the graph.
Since f (x) is not true for the second point, repeat the first step. The graph has one branch below the x-axis and one
branch above the x-axis. This indicates that n is odd. Assume n = 3.
When n =3,
for the point . If this power function is true for the second point
,
then f (x) can represent the graph.
Since f (x) is true for
,
is a power function for the graph.
79. OPTICS A contact lens with the appropriate depth ensures proper fit and oxygen permeation. The depth of a lens
can be calculated using the formula
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the diameter, with all units in millimeters.
, where S is the depth, r is the radius of curvature, and d is
Page 49
Since f (x) is true for
,
is a power function for the graph.
2-1 Power and Radical Functions
79. OPTICS A contact lens with the appropriate depth ensures proper fit and oxygen permeation. The depth of a lens
can be calculated using the formula
, where S is the depth, r is the radius of curvature, and d is
the diameter, with all units in millimeters.
a. If the depth of the contact lens is 1.15 millimeters and the radius of curvature is 7.50 millimeters, what is the
diameter of the contact lens?
b. If the depth of the contact lens is increased by 0.1 millimeter and the diameter of the lens is 8.2 millimeters, what
radius of curvature would be required?
c. If the radius of curvature remains constant, does the depth of the contact lens increase or decrease as the
diameter increases?
SOLUTION: a. Substitute S = 1.15 and r = 7.50 into
and solve for d.
Since the each side of the equation was raised to a power, check the solution in the original equation.
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Page 50
2-1 Power and Radical Functions
Since the each side of the equation was raised to a power, check the solution in the original equation.
The diameter of the contact is 7.98 mm.
b. Substitute S = 1.15 + 0.1 or 1.25 and d = 8.2 into
and solve for r.
Since the each side of the equation was raised to a power, check the solution in the original equation.
The radius of curvature required is 7.35 mm.
c. As d increases, the value of
increases. Thus, the value of the radicand, the radicand decreases, and assuming that it remains nonnegative,
difference found by
, will also decrease. As
decreases. Therefore, the will increase, creating greater values for the depth S.
80. MULTIPLE REPRESENTATIONS In this problem, you will investigate the average rates of change of power
functions.
n
a. GRAPHICAL For power functions of the form f (x) = x , graph a function for two values of n such that 0 < n
< 1, n = 1, and two values of n such that n > 1.
b. TABULAR Copy and complete the table, using your graphs from part a to analyze the average rates of
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change
the functions
as x approaches infinity. Describe this rate as increasing, constant, or decreasing. Page 51
80. MULTIPLE REPRESENTATIONS In this problem, you will investigate the average rates of change of power
functions.
2-1 Power
and Radical Functions
n
a. GRAPHICAL For power functions of the form f (x) = x , graph a function for two values of n such that 0 < n
< 1, n = 1, and two values of n such that n > 1.
b. TABULAR Copy and complete the table, using your graphs from part a to analyze the average rates of
change of the functions as x approaches infinity. Describe this rate as increasing, constant, or decreasing.
c. VERBAL Make a conjecture about the average rate of change of a power function as x approaches infinity
for the intervals 0 < n < 1, n = 1, and n > 1.
SOLUTION: a. Sample answer:
b.
c. Sample answer: When 0 < n < 1, the average rate of change of the function decreases as x approaches infinity.
When n = 1, the average rate of change of the function is constant as x approaches infinity. When n > 1, the
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Page 52
average rate of change of the function increases as x approaches infinity.
2-1 Power and Radical Functions
c. Sample answer: When 0 < n < 1, the average rate of change of the function decreases as x approaches infinity.
When n = 1, the average rate of change of the function is constant as x approaches infinity. When n > 1, the
average rate of change of the function increases as x approaches infinity.
81. CHALLENGE Show that
.
SOLUTION: 82. REASONING Consider y = 2x.
a. Describe the value of y if x < 0.
b. Describe the value of y if 0 < x < 1.
c. Describe the value of y if x > 1.
d. Write a conjecture about the relationship between the values of the base and power if the exponent is greater
than or less than 1. Justify your answer.
SOLUTION: 0
a. For x < 0, analyze the function for x = 0. When x = 0, y = 2 or 1. So, if x < 0, y < 1. A power of a positive
number is never negative. So, y > 0. This results in values for y of 0 < y < 1.
b. For 0 < x < 1, analyze the function for x = 0 and x = 1. When x = 0, y = 20 or 1. When x = 1, y = 21 or 2. Thus,
if x is 0 < x < 1, then y is 1< y < 2.
1
c. For x > 1, analyze the function for x = 1. When x = 1, y = 2 or 2. For greater values of x, y will be greater than
2 or y > 2.
d. Sample answer: If the exponent is less than 0, the power is greater than 0 and less than 1. If the exponent is
greater than 0 and less than 1, the power is greater than 1 and less than the base. If the exponent is greater than 1,
the power is greater than the base. Any number to the zero power is 1. Thus, if the exponent is less than 0, the
power is less than 1. A power of a positive number is never negative, so the power is greater than 0. Any number
to the zero power is 1 and to the first power is itself. Thus, if the exponent is greater than 0 and less than 1, the
power is between 1 and the base. Any number to the first power is itself. Thus, if the exponent is greater than 1,
the power is greater than the base.
83. PREWRITE Your Senior Project is to tutor an underclassman for four tutoring sessions on power and radical
functions. Make a plan for writing that addresses purpose, audience, a controlling idea, logical sequence, and time
frame for completion.
SOLUTION: eSolutions Manual - Powered by Cognero
Sample answer:
Purpose: To ultimately help an underclassman better understand what power functions are and how they are
Page 53
2-1
the power is greater than the base. Any number to the zero power is 1. Thus, if the exponent is less than 0, the
power is less than 1. A power of a positive number is never negative, so the power is greater than 0. Any number
to the zero power is 1 and to the first power is itself. Thus, if the exponent is greater than 0 and less than 1, the
power isand
between
1 and the
base. Any number to the first power is itself. Thus, if the exponent is greater than 1,
Power
Radical
Functions
the power is greater than the base.
83. PREWRITE Your Senior Project is to tutor an underclassman for four tutoring sessions on power and radical
functions. Make a plan for writing that addresses purpose, audience, a controlling idea, logical sequence, and time
frame for completion.
SOLUTION: Sample answer:
Purpose: To ultimately help an underclassman better understand what power functions are and how they are
related to radical functions.
Audience: Either an Algebra 1 or Algebra 2 student.
Controlling idea: Understand the relationship between power and radical functions and how they can be used to
solve equations.
Logical sequence and timeframe:
st
1 session: Practice recognizing power functions.
nd
2 session: Practice graphing power functions.
rd
3 session: Practice writing and graphing exponential functions in radical form.
th
4 session: Practice solving equations involving radicals.
84. REASONING Given
, where a and b are integers with no common factors, determine whether each
statement is true or false . Explain.
a. If the value of b is even and the value of a is odd, then the function is undefined for x < 0.
b. If the value of a is even and the value of b is odd, then the function is undefined for x < 0.
c. If the value of a is 1, then the function is defined for all x.
SOLUTION: a. True; sample answer: f (x) can be written as
a negative number is undefined.
. If b is even and a is odd, then x ≥ 0. An even root of b. False; sample answer: f (x) can be written as
. If b is odd and a is even, then f (x) is defined for all x.
c. False; sample answer: f (x) can be written as
then x ≥ 0.
. If b is odd, then f (x) is defined for all x. If b is even,
85. REASONING Consider
. How would you expect the graph of the function to change as n
increases if n is odd and greater than or equal to 3?
SOLUTION: Sample answer: As n increases, the value of
approaches 0. This means that the value of will approach 1 when x is positive and −1 when x is negative. Therefore, for positive values of x, f (x) will approach 1 + 5 or 6 and
will resemble the line y = 6. For negative values of x, f (x) will approach −1 + 5 or 4 and will resemble the line y =
4.
86. Writing in Math Use words, graphs, tables, and equations to show the relationship between functions in
exponential form and in radical form.
SOLUTION: See students' work.
87. FINANCE If you deposit $1000 in a savings account with an interest rate of r compounded annually, then the
balance
the account
after
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3
3 years is given by B(r) = 1000(1 + r) , where r is written as a decimal.
a. Find a formula for the interest rate r required to achieve a balance of B in the account after 3 years.
b. What interest rate will yield a balance of $1100 after 3 years?
Page 54
86. Writing in Math Use words, graphs, tables, and equations to show the relationship between functions in
exponential form and in radical form.
SOLUTION: 2-1 Power
and Radical Functions
See students' work.
87. FINANCE If you deposit $1000 in a savings account with an interest rate of r compounded annually, then the
3
balance in the account after 3 years is given by B(r) = 1000(1 + r) , where r is written as a decimal.
a. Find a formula for the interest rate r required to achieve a balance of B in the account after 3 years.
b. What interest rate will yield a balance of $1100 after 3 years?
SOLUTION: 3
a. Solve B(r) = 1000(1 + r) for r. Substitute B for B(r).
The formula for the interest rate r required is
.
b. Substitute B = 1100 into the equation found in part a.
The interest rate needed is 3.23%.
Find (f + g)(x), (f – g )(x), (f ⋅ g)(x), and
for each f (x) and g(x). State the domain of each new
function.
88. f (x) = x2 – 2x
g(x) = x + 9
SOLUTION: There are no restrictions on the domain. So, D = (−
,
).
There are no restrictions on the domain. So, D = (−
,
).
There are no restrictions on the domain. So, D = (−
,
).
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The denominator cannot equal zero. So,
Page 55
.
2-1 Power and Radical Functions
The interest rate needed is 3.23%.
Find (f + g)(x), (f – g )(x), (f ⋅ g)(x), and
for each f (x) and g(x). State the domain of each new
function.
88. f (x) = x2 – 2x
g(x) = x + 9
SOLUTION: There are no restrictions on the domain. So, D = (−
,
).
There are no restrictions on the domain. So, D = (−
,
).
There are no restrictions on the domain. So, D = (−
,
).
The denominator cannot equal zero. So,
.
89. f (x) =
2
g(x) = x – 1
SOLUTION: The denominator cannot equal zero. So,
.
The denominator cannot equal zero. So,
.
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Page 56
There are no restrictions on the domain. So, D = (−
2-1 Power
and Radical
Functions
The denominator
cannot equal
zero. So,
,
).
.
89. f (x) =
2
g(x) = x – 1
SOLUTION: The denominator cannot equal zero. So,
.
The denominator cannot equal zero. So,
.
The denominator of f (x) cannot equal zero. So,
The denominator will equal zero when x = ±1. So,
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90. f (x) =
2
g(x) = x + 5x
.
.
Page 57
2-1 Power and Radical Functions
The denominator will equal zero when x = ±1. So,
.
90. f (x) =
2
g(x) = x + 5x
SOLUTION: The denominator cannot equal zero. So,
.
The denominator cannot equal zero. So,
.
The denominator cannot equal zero. So,
.
2
Since x + 5x is a factor of the denominator, the denominator will equal zero for x = 0 and x = −5, in addition to x =
7. So,
.
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Use
the graph
ofbyf (x)
to graph g(x)
91. f (x) = −4x + 2
= |f (x)| and h(x) = f (|x|).
Page 58
2
Since x and
+ 5x isRadical
a factor ofFunctions
the denominator, the denominator will equal zero for x = 0 and x = −5, in addition to x =
2-1 Power
7. So,
.
Use the graph of f (x) to graph g(x) = |f (x)| and h(x) = f (|x|).
91. f (x) = −4x + 2
SOLUTION: Evaluate the function for several x-values in its domain.
x
0
1
2
3
−3
−2
−1
f (x)
14
10
6
2
−2
−6
−10
Use these points to construct a graph.
g(x) is the graph of the absolute values of f (x). Evaluate g(x) for several x-values in its domain.
x
0
1
2
3
−3
−2
−1
g(x)
14
10
6
2
2
6
10
Use these points to construct a graph.
h(x) is the graph of f (x) for the absolute values of x. Evaluate h(x) for several x-values in its domain.
x
0
1
2
3
−3
−2
−1
h(x)
2
−10 −6
−2
−2 −6 −10
Use these points to construct a graph.
92. f (x) =
−6
SOLUTION: Evaluate the function for several x-values in its domain.
x
0
1
4
−3
−2
−1
eSolutions
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Cognero
f (x)
−6
−4.6 −4.3
−4
−3.4
Use these points to construct a graph.
6
−3
Page 59
2-1 Power and Radical Functions
92. f (x) =
−6
SOLUTION: Evaluate the function for several x-values in its domain.
x
0
1
4
−3
−2
−1
f (x)
−6
−5
−4.6 −4.3
−4
−3.4
Use these points to construct a graph.
6
−3
g(x) is the graph of the absolute values of f (x). Evaluate g(x) for several x-values in its domain.
x
0
1
4
6
−3
−2
−1
g(x)
6
5
4.6 4.3
4
3.4
3
Use these points to construct a graph.
h(x) is the graph of f (x) for the absolute values of x. Evaluate h(x) for several x-values in its domain.
x
0
1
4
6
−6
−4
−1
h(x)
−3 −3.4 −4 −4.3 −4 −3.4 −3
Use these points to construct a graph.
93. f (x) = x2 − 3x − 10
SOLUTION: Evaluate the function for several x-values in its domain.
x
0
1
2
5
−3
−2
f (x)
8
0
0
−10
−12
−12
Use these points to construct a graph.
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6
8
Page 60
2-1 Power and Radical Functions
93. f (x) = x2 − 3x − 10
SOLUTION: Evaluate the function for several x-values in its domain.
x
0
1
2
5
−3
−2
f (x)
8
0
0
−10
−12
−12
Use these points to construct a graph.
6
8
g(x) is the graph of the absolute values of f (x). Evaluate g(x) for several x-values in its domain.
x
0
1
2
5
6
−3
−2
g(x)
8
0
10
12
12
0
8
Use these points to construct a graph.
h(x) is the graph of f (x) for the absolute values of x. Evaluate h(x) for several x-values in its domain.
x
0
1
2
3
−3
−2
−1
h(x)
−10 −12 −12 −10 −12 −12 −10
Use these points to construct a graph.
Use the graph of each function to estimate intervals to the nearest 0.5 unit on which the function is
increasing, decreasing, or constant. Support the answer numerically.
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94. Page 61
2-1 Power and Radical Functions
Use the graph of each function to estimate intervals to the nearest 0.5 unit on which the function is
increasing, decreasing, or constant. Support the answer numerically.
94. SOLUTION: From the graph, we can estimate that f is increasing on (− , −3), decreasing on (−3, 1), and increasing on (1, ∞).
Create a table of values using x-values in each interval.
(− , −3)
x
−11
−9
−7
−5
−3
f (x)
5
−455 −220 −81 −14
(−3, 1)
x
0
1
−3
−2
−1
f (x)
5
4
0
−4
−5
(1, ∞)
x
1
3
5
7
9
f (x)
14
81
220 455
−5
The tables support that f is increasing on (− , −3), decreasing on (−3, 1), and increasing on (1, ).
95. SOLUTION: From the graph, we can estimate that f is increasing on (− , −4) and increasing on (−4,
values using x-values in each interval.
(− , −4)
x
−12 −10
−8
−6 −4.1
f (x)
1.86 2.17 2.75 4.5
71
(−4, )
x
0
2
4
−3.9
−2
f (x)
−69 −2.5 −0.75 −0.17 0.13
The tables support that f is increasing on (− , −4) and increasing on (−4, ).
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96. SOLUTION: ). Create a table of
Page 62
2-1
1.86 2.17 2.75 4.5
71
(−4, )
x
0
2
4
−3.9
−2
f (x)
0.13
−69
−2.5
−0.75
−0.17
Power and Radical Functions
The tables support that f is increasing on (−
, −4) and increasing on (−4,
).
96. SOLUTION: From the graph, we can estimate that f is increasing on (− , −3.5), decreasing on (−3.5, −2), decreasing on (−2,
0), and increasing on (0, ). Create a table of values using x-values in each interval.
(− , −3.5)
x
−11
−9
−7
−5 −3.5
f (x)
−13.3 −11.4 −9.6 −8 −7.5
(−3.5, −2)
x
−3.5
−3 −2.5 −2.25 −2.1
f (x)
−7.5
−8 −10.5 −16.25 −34.1
(−2, 0)
x
0
−1.9 −1.5
−1 −0.5
f (x)
26.1 2.5
0
−0.5 −0.5
(0, )
x
0
2
4
6
8
f (x)
2.5
4.4
6.3
−0.5 0.8
The tables support that f is increasing on (− , −3.5), decreasing on (−3.5, −2), decreasing on (−2, 0), and
increasing on (0, ).
Simplify.
97. SOLUTION: eSolutions
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SOLUTION: Page 63
2-1 Power and Radical Functions
98. SOLUTION: 99. SOLUTION: 100. SAT/ACT If m and n are both positive, then which of the following is equivalent to
?
A 3m
B 6m
C4
D6
E8
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SOLUTION: Page 64
2-1 Power and Radical Functions
100. SAT/ACT If m and n are both positive, then which of the following is equivalent to
?
A 3m
B 6m
C4
D6
E8
SOLUTION: The correct answer is D.
101. REVIEW If f (x, y) = x2y 3 and f (a, b) = 10, what is the value of f (2a, 2b)?
F 50
G 100
H 160
J 320
SOLUTION: Evaluate f (2a, 2b).
2 3
Substitute a b = 10.
The correct answer is J.
102. REVIEW The number of minutes m it takes c children to eat p pieces of pizza varies directly as the number of
pieces of pizza and inversely as the number of children. If it takes 5 children 30 minutes to eat 10 pieces of pizza,
how many minutes should it take 15 children to eat 50 pieces of pizza?
A 30
B 40
C 50
D 60
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SOLUTION: Write an equation to model the situation using m, c, p , and a constant k.
2-1 Power and Radical Functions
The correct answer is J.
102. REVIEW The number of minutes m it takes c children to eat p pieces of pizza varies directly as the number of
pieces of pizza and inversely as the number of children. If it takes 5 children 30 minutes to eat 10 pieces of pizza,
how many minutes should it take 15 children to eat 50 pieces of pizza?
A 30
B 40
C 50
D 60
SOLUTION: Write an equation to model the situation using m, c, p , and a constant k.
m=
Substitute c = 5, m = 30, and p = 10 to solve for k.
Substitute k = 15, c = 15, and p = 50 to find m.
The correct answer is C.
103. If
, then m = ?
F3
G4
H5
J6
SOLUTION: Since the each side of the equation was raised to a power, check the solution in the original equation.
The correct answer is H.
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2-1 Power and Radical Functions
The correct answer is C.
103. If
, then m = ?
F3
G4
H5
J6
SOLUTION: Since the each side of the equation was raised to a power, check the solution in the original equation.
The correct answer is H.
eSolutions Manual - Powered by Cognero
Page 67