Chapter 4: Rational, Power, and Root Functions
4.1:
Rational Functions and Gr
aphs
1.
The only value for
x
that cannot be used as input is 0.
The domain is
.
It is not possible for this function to output the value 0.
The range is
.
2.
The only value for
x
that cannot be used as input is 0.
The domain is
.
The function will output only positive values.
The range is
.
3.
The function decreases everywhere it is defined,
.
It never increases and is never constant.
4.
The function increases on
.
It decreases on
.
The function is never constant.
5.
Because the function is undefined when
, the vertical asymptote has the equation
.
As
increases with out bound, the graph of the function will move closer and closer to the graph of
.
6.
Because the function is undefined when
, the vertical asymptote has the equation
.
As
increases with out bound, the graph of the function will move closer and closer to the graph of
.
7.
Because
, the function is even.
The graph has symmetry with respect to the
y
axis.
8.
Because
, the function is odd.
The graph has symmetry with respect to the origin.
9.
Graphs A, B, and C have domain
because each has a vertical asymptote at
.
10.
Graph B has range
because it exists above and below the horizontal asymptote at
.
11.
Graph A has range
because it exists above and below the horizontal asymptote at
.
12.
Graphs C and D have range
because each exists only above the horizontal asymptote at
.
13.
The only graph that would intersect the line
exactly one time is graph A.
14.
Because graph A exists above and below the horizontal asymptote
, its range is
.
15.
Graphs A, C, and D have the
x
axis as a horizontal asymptote.
16.
Noting that graph D has a hole, graph C is the only graph that is symmetric with respect to a vertical line.
17.
Window C gives the most accurate depiction of the graph.
See Figures 17a, 17b and 17c.
Figure 17a
Figure 17b
Figure 17c
18.
Window A gives the most accurate depiction of the graph.
See Figures 18a, 18b and 18c.
Figure 18a
Figure 18b
Figure 18c
Yscl
1
Xscl
1
Yscl
1
Xscl
1
Yscl
1
Xscl
1
3
4.7, 4.7
4
by
3
3.1, 3.1
4
3
4.7, 4.7
4
by
3
6.2, 6.2
4
3
4.7, 4.7
4
by
3
0, 12.4
4
Yscl
1
Xscl
1
Yscl
1
Xscl
1
Yscl
1
Xscl
1
3
9.4, 9.4
4
by
3
3.1, 3.1
4
3
14.4, 4.4
4
by
3
0, 5
4
3
4.7, 4.7
4
by
3
3.1, 3.1
4
1
q
, 0
2
h
1
0,
q
2
y
0
y
3
y
0
1
0,
q
2
y
0
1
q
, 0
2
h
1
0,
q
2
y
3
1
q
, 3
2
h
1
3,
q
2
x
3
1
q
, 3
2
h
1
3,
q
2
ƒ
1
x
2
ƒ
1
x
2
ƒ
1
x
2
ƒ
1
x
2
y
4
0
x
0
x
2
x
2
y
2
0
x
0
x
3
x
3
1
0,
q
2
1
q
, 0
2
1
q
, 0
2
h
1
0,
q
2
1
0,
q
2
1
q
, 0
2
h
1
0,
q
2
1
q
, 0
2
h
1
0,
q
2
1
q
, 0
2
h
1
0,
q
2
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19. Let
, then
.
To obtain the graph of
f
, stretch the graph of
vertically by a factor of 2.
See Figures 19a and 19b.
The domain is
.
The range is
.
Figure 19a
Figure 19b
20. Let
, then
.
To obtain the graph of
f
, stretch the graph of
vertically by a factor of 3 and reflect it across the
x
axis
or
the
y
axis.
See Figures 20a and 20b.
The domain is
.
The range is
.
Figure 20a
Figure 20b
21. Let
, then
.
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 Spring '08
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 Calculus, Asymptotes, Rational Functions, Fraction, Rational function

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