Graph and analyze each function. Describe the
domain, range, intercepts, end behavior,
continuity, and where the function is increasing
or decreasing.
2-1 Power and Radical Functions
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.
19. SOLUTION: Evaluate the function for several x-values in its
domain.
x
−6
−4
−2
0
2
4
6
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.
r
0.5
1
1.5
2
2.5
3
3.5
V(r)
0.5
4.2
14.1
33.5
65.5
113.1
179.6
f(x)
8.6
7.9
6.9
0
−6.9
−7.9
−8.6
Use these points to construct a graph.
Use these points to construct a graph.
All values of x are included in the graph, so the
function exists for all values of x, and D = (−∞, ∞).
All values of y are included in the graph, so the
function exists for all values of y, and R = (−∞, ∞).
Graph and analyze each function. Describe the
domain, range, intercepts, end behavior,
continuity, and where the function is increasing
or decreasing.
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
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 (−∞, ∞).
33. WEATHER The wind chill temperature is the eSolutions
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The only time the graph intersects the axes is when
it goes through the origin, so the x- and y-intercepts
are both 0.
The y-value approaches positive infinity as x
approaches negative infinity, and negative infinity as
x approaches positive infinity, so
and
.
a graph.
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
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various speeds when the actual temperature is 50ºF.
graph is decreasing on (−∞, ∞).
33. WEATHER
2-1
Power and The wind chill temperature is the Radical Functions
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.
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
value function from the CALC menuon the
graphing calculator. Let x = 65.
The wind chill temperature when the wind speed is
65 miles per hour is about 39.54°F
42. FLUID MECHANICS The velocity of the water 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.
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.
P
0
1
2
3
4
5
6
b. Use the power regression function on the
graphing calculator to find values for a and b.
V(P)
0
12.1
17.1
21
24.2
27.1
29.7
Use these points to construct a graph.
−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
value function from the CALC menuon the
graphing calculator. Let x = 65.
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The wind chill temperature when the wind speed is
65 miles per hour is about 39.54°F
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, )
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The velocities include nonnegative values, so R = [0,
)
is continuous on [0,
2-1
).
As you read the graph from left to right, it is going up
from 0 to positive infinity, so the graph is increasing
on [0, ∞).
b. Since it is an even-degree radical function, the
Power and Radical Functions
domain is restricted to nonnegative values for the
radicand, P. Thus, P ≥ 0.
D = [0, )
The velocities include nonnegative values, so R = [0,
)
Solve each equation.
46. −3 =
− SOLUTION: The velocity approaches infinity as pressure
.
approaches infinity, so There are no breaks, holes, or gaps in the graph, so it
is continuous on [0, ).
Now isolate the remaining radical.
As you read the graph from left to right, it is going up
from 0 to positive infinity, so the graph is increasing
on [0, ∞).
Solve each equation.
46. −3 =
− SOLUTION: Now isolate the remaining radical.
Since the each side of the equation was raised to a
power, check the solutions in the original equation.
x = 13
x= 1.75
Since the each side of the equation was raised to a
power, check the solutions in the original equation.
x = 13
One solution checks and the other solution does not.
Therefore, the solution is x = 13.
Solve each inequality.
66. SOLUTION: eSolutions Manual - Powered by Cognero
x= 1.75
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One solution checks and the other solution does not.
2-1 Therefore,
Power and
Radical
the solution
is x =Functions
13.
Solve each inequality.
66. 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 = 24.
Since the root 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.
Therefore, the solution is −2 ≤ x ≤ 25.
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