 # Graph and analyze each function. Describe the

by user

on
22

views

Report

#### Transcript

Graph and analyze each function. Describe the
``` As you read the graph from left to right, it is going
down from negative infinity to 0, and then going up
from 0 to positive infinity, so the graph is decreasing
on (− , 0) and increasing on (0, ).
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
3. h(x) = −x3
SOLUTION: Evaluate the function for several x-values in its
domain.
x
−3
−2
−1
0
1
2
3
SOLUTION: Evaluate the function for several x-values in its
domain.
x
−3
−2
−1
0
1
2
3
f(x)
45
20
5
0
5
20
45
h(x)
27
8
1
0
−1
−8
−27
Use these points to construct a graph.
Use these points to construct a graph.
The function is a monomial with an even degree and
a positive value for a.
All values of x are included in the graph, so the
function exists for all values of x, and D = (− ,
).
The function is a monomial with an odd degree and a
negative value for a.
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 = (− ∞, ∞).
The only values of y that are included in the graph
are 0 through infinity, so R = [0, ).
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-values approach positive infinity as x
approaches negative or positive infinity, so
(x) = ∞ and f (x) = ∞.
f
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 0, and then going up
from 0 to positive infinity, so the graph is decreasing
on (− , 0) and increasing on (0, ).
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
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 (− ∞, ∞).
5. g(x) =
x
9
3. h(x) = −x3
SOLUTION: Page 1
SOLUTION: Evaluate the function for several x-values in its
is continuous for all real numbers.
As you read the graph from left to right, it is going
down
fromand
negative
infinity Functions
to positive infinity, so the
Power
graph is decreasing on (− ∞, ∞).
As you read the graph from left to right, it is going up
from negative infinity to positive infinity so the graph
is increasing on (− ∞, ∞).
2-1
it is continuous for all real numbers.
5. g(x) =
x
9
7. SOLUTION: SOLUTION: Evaluate the function for several x-values in its
domain.
Evaluate the function for several x-values in its
domain.
x
−3
−2
−1
0
1
2
3
x
−3
−2
−1
0
1
2
3
g(x)
−6561
−170.7
−0.3
0
0.3
170.7
6561
Use these points to construct a graph.
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
positive value for a.
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 = (− ∞, ∞).
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 positive infinity, and negative infinity as x
approaches negative infinity, so
and
.
The function is a monomial with an odd degree and a
negative value for a.
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 = (− ∞, ∞).
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
There are no breaks, holes, or gaps in the graph, so it
is continuous for all real numbers.
and
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 up
from negative infinity to positive infinity so the graph
is increasing on (− ∞, ∞).
7. eSolutions
SOLUTION: Evaluate the function for several x-values in its
domain.
.
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 (− ∞, ∞).
9. f (x) = 2x −4
SOLUTION: Evaluate the function for several x-values in its
Page 2
2-1
is continuous for all real numbers.
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going down from
0 to positive infinity, so the graph is increasing on (− ∞, 0) and decreasing on (0, ∞).
As you read the graph from left to right, it is going
down
fromand
negative
infinity Functions
to positive infinity, so the
Power
graph is decreasing on (− ∞, ∞).
9. f (x) = 2x −4
11. f (x) = −8x −5
SOLUTION: SOLUTION: Evaluate the function for several x-values in its
domain.
Evaluate the function for several x-values in its
domain.
x
−3
−2
−1
0
1
2
3
x
−3
−2
−1
0
1
2
3
f(x)
0.025
0.125
2
undefined
2
0.125
0.025
f(x)
0.03
0.25
8
0
−8
−0.25
−0.03
Use these points to construct a graph.
Use these points to construct a graph.
Since the power is negative, the function will be
undefined at x = 0 and D = (− ∞, 0)∪(0, ∞).
Since the power is negative, the function will be
undefined at x = 0, and D = (− ∞, 0)∪(0, ∞).
The only values of y that are included in the graph
are greater than 0 through infinity, so R = (0, ∞).
All values of y are included on the graph except 0, so
R = (− ∞, 0)∪(0, ∞).
The graph never intersects either axis, so there are
no x- or y-intercepts.
The graph never intersects either axis, so there are
no x- or y-intercepts.
The y-values approach zero as x approaches
negative or positive infinity, so
The y-values approach zero as x approaches
negative or positive infinity, so
and
and
.
The graph has an infinite discontinuity at x = 0.
The graph has an infinite discontinuity at x = 0.
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going down from
0 to positive infinity, so the graph is increasing on (− ∞, 0) and decreasing on (0, ∞).
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going up again
from 0 to positive infinity, so the graph is increasing
on (− ∞, 0)∪(0, ∞).
11. f (x) = −8x −5
SOLUTION: Evaluate the function for several x-values in its
domain.
x
f(x)
0.03
−3
0.25
−2
8
−1
0
0
13. SOLUTION: Evaluate the function for several x-values in its
domain.
x
−1.5
−1
f(x)
0.01
0.4
Page 3
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going up again
from
0 to positive
infinity, soFunctions
the graph is increasing
Power
on (− ∞, 0)∪(0, ∞).
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going up again
from 0 to positive infinity, so the graph is increasing
on (− ∞, 0)∪(0, ∞).
2-1
13. 15. h(x) =
x
−3
SOLUTION: SOLUTION: Evaluate the function for several x-values in its
domain.
Evaluate the function for several x-values in its
domain.
x
−1.5
−1
−0.5
0
f(x)
0.01
0.4
204.8
0
0.5
1
1.5
−204.8
−0.4
−0.01
Use these points to construct a graph.
x
−1.5
−1
−0.5
0
0.5
h(x)
−0.22
−0.75
−6
0
6
1
1.5
0.75
0.22
Use these points to construct a graph.
Since the power is negative, the function will be
undefined at x = 0, and D = (− ∞, 0)∪(0, ∞).
All values of y are included in the graph except 0, so
R = (− ∞, 0)∪(0, ∞).
The graph never intersects either axis, so there are
no x- and y-intercepts.
The y-values approach 0 as x approaches negative
or positive infinity, so
and
.
Since the power is negative, the function will be
undefined at x = 0, and D = (− ∞, 0)∪(0, ∞).
All values of y are included on the graph except 0, so
R = (− ∞, 0)∪(0, ∞).
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-values approach zero as x approaches
negative or positive infinity, so
The graph has an infinite discontinuity at x = 0.
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going up again
from 0 to positive infinity, so the graph is increasing
on (− ∞, 0)∪(0, ∞).
15. h(x) =
x
−3
SOLUTION: Evaluate the function for several x-values in its
domain.
x
h(x)by Cognero
eSolutions Manual
- Powered
−1.5
−1
−0.22
−0.75
and
.
The graph has an infinite discontinuity at x = 0.
As you read the graph from left to right, it is going
down from negative infinity to 0, and then going
down again from 0 to positive infinity, so the graph is
decreasing on (− ∞, 0)∪(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: Page 4
2-1
As you read the graph from left to right, it is going
down from negative infinity to 0, and then going
down
again
0 to positive
infinity, so the graph is
Power
and
Functions
decreasing on (− ∞, 0)∪(0, ∞).
17. GEOMETRY The volume of a sphere is given by V(r) =
3
Graph and analyze each function. Describe the
domain, range, intercepts, end behavior,
continuity, and where the function is increasing
or decreasing.
19. πr , where r is the radius.
SOLUTION: a. State the domain and range of the function.
b. Graph the function.
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 = (−∞, ∞).
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
.
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
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 (−∞, ∞).
21. SOLUTION: eSolutions
Powered
Cognero
UseManual
these -points
tobyconstruct
There are no breaks, holes, or gaps in the graph, so it
is continuous for all real numbers.
a graph.
Evaluate the function for several x-values in its
domain.
x
f(x)
Page 5
is continuous over the domain (0, ∞).
2-1
As you read the graph from left to right, it is going
down from negative infinity to positive infinity, so the
Power
Functions
graph
is decreasing
on (−∞, ∞).
21. As you read the graph from left to right, it is going
down from 0 to positive infinity, so the graph is
decreasing on (0, ∞).
23. SOLUTION: SOLUTION: Evaluate the function for several x-values in its
domain.
x
1
2
3
4
5
6
7
Evaluate the function for several x-values in its
domain.
f(x)
10
8.9
8.3
7.9
7.6
7.4
7.2
x
−3
−2
−1
0
1
2
3
Use these points to construct a graph.
Use these points to construct a graph.
The exponent is positive and has an odd
denominator, so there are no restrictions on the
domain. D = (−∞, ∞).
Since the denominator of the power is even and the
power is negative, the domain must be restricted to
positive values, D = (0, ∞).
All values of y are included in the graph, so the
function exists for all values of y, and R = (−∞, ∞).
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 negative infinity as x
approaches negative infinity, and approaches positive
infinity as x approaches positive infinity, so
and .
The only values of y that are included in the graph
are greater than 0 through infinity, so R = (0, ∞).
The graph never intersects either axis, so there are
no x- or y-intercepts.
The y-values approach zero as x approaches positive
infinity, so
.
There are no breaks, holes, or gaps in the graph, so it
is continuous over the domain (0, ∞).
As you read the graph from left to right, it is going
down from 0 to positive infinity, so the graph is
decreasing on (0, ∞).
23. SOLUTION: Evaluate the function for several x-values in its
domain.
eSolutions Manual
- Powered
x
h(x)by Cognero
−3
−2
−1.45
−1.13
h(x)
−1.45
−1.13
−0.75
0
0.75
1.13
1.45
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
0
1
0.63
0.48
Page 6
2-1
The y-value approaches negative infinity as x
approaches negative infinity, and approaches positive
infinity as x approaches positive infinity, so
and Functions.
25. As you read the graph from left to right, it is going up
from negative infinity to 0, and then going down from
0 to positive infinity, so the graph is increasing on
(−∞, 0) and decreasing on (0, ∞).
27. SOLUTION: SOLUTION: Evaluate the function for several x-values in its
domain.
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
0
1
0.63
0.48
x
0
0.5
1
1.5
2
2.5
3
h(x)
0
−1.2
−4
−8.1
−13.5
−19.9
−27.4
Use these points to construct a graph.
Use these points to construct a graph.
Since the power is negative, the function will be
undefined at x = 0, and D = (−∞, 0)∪(0, ∞).
Since the denominator of the power is even, the
domain must be restricted to nonnegative values. D
= [0, ∞).
The only values of y that are included in the graph
are greater than 0 through infinity, so R = (0, ∞).
The only values of y that are included in the graph
are negative infinity through 0, so R = (–∞, 0].
The graph never intersects either axis, so there are
no x- or y-intercepts.
The y-values approach zero as x approaches
negative or positive infinity, so
and
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 negative infinity as x
approaches positive infinity, so
.
There are no breaks, holes, or gaps in the graph, so it is continuous over the domain [0, ∞)
As you read the graph from left to right, it is going
down from 0 to positive infinity, so the graph is
decreasing on (0, ∞).
.
The graph has an infinite discontinuity at x = 0.
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going down from
0 to positive infinity, so the graph is increasing on
(−∞, 0) and decreasing on (0, ∞).
27. 29. SOLUTION: Evaluate the function for several x-values in its
domain.
x
h(x)by Cognero
eSolutions Manual
- Powered
0
0.5
0
−1.2
SOLUTION: Evaluate the function for several x-values in its
domain.
x
−3
h(x)
0.11
Page 7
2-1
it is continuous over the domain [0, ∞)
As you read the graph from left to right, it is going
down
fromand
0 to positive
infinity,
so the graph is
Power
Functions
decreasing on (0, ∞).
29. SOLUTION: Evaluate the function for several x-values in its
domain.
x
−3
−2
−1
0
1
2
3
h(x)
0.11
0.22
0.67
0
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, ∞).
The only values of y that are included in the graph
are greater than 0 through infinity, so R = (0, ∞).
The graph never intersects either axis, so there are
to x- or y-intercepts.
The y-values approach 0 as x approaches negative
or positive infinity, so
and
.
There is an infinite discontinuity at x = 0.
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going down from
0 to positive infinity, so the graph is increasing on
(−∞, 0) and decreasing on (0, ∞).