Part II: The Answers
158
Answers
1–100
80.
domain: −∞ −
(
)∪ ∞
[ )
, 1
2, ; range: − −
(
)∪ [ )
3 1
2 4
,
,
Notice that the function is continuous on its domain.
For the portion of the graph that’s to the left of the y-axis, there’s no largest x value
due to the hole at (–1, –1); instead, the x values get arbitrarily close to –1. The graph
then extends indefinitely to the left so that the domain for the left side of the graph is
(–∞, –1). There’s also no largest y value due to the hole at (–1, –1); the y values get
arbitrarily close to –1. The graph then decreases as it approaches the horizontal
asymptote at y = –3, so the range for this portion of the graph is (–3, –1).
For the portion of the graph that’s to the right of the y-axis, the smallest x value is at
the point (2, 2). The graph then extends indefinitely to the right, so the domain for the
right side of the graph is [2, ∞). The smallest y value also occurs at the point (2, 2), and
then the graph increases as it approaches the horizontal asymptote at y = 4; therefore,
the range of this portion of the graph is [2, 4).
Putting the two parts together, the domain of the function is −∞ −
(
)∪ ∞
[ )
, 1
2, , and the
range is − −
(
)∪ [ )
3 1
2 4
,
, .
81.
lim
x
x
x
→−∞
−
+
(
) = ∞
3
40
33
6
5
, lim x
x
x
→∞
−
+
(
) = ∞
3
40
33
6
5
To determine the end behavior of a polynomial, you just have to determine the end
behavior of the highest-powered term.
As x approaches –∞, you have
lim
lim
x
x
x
x
x
→−
→
∞
− ∞
−
+
(
) = ( )
= ∞
3
40
33
3
6
5
6
Note that even though the limit is approaching negative infinity, the limit is still positive due to the even exponent.
As x approaches ∞, you have
lim
lim x
x
x
x
x
→∞
→∞
−
+
(
) = ( )
= ∞
3
40
33
3
6
5
6
82.
lim
x
x
x
x
x
→−∞
−
+
−
+
− )
(
= ∞
7
33
51
19
1
9
8
7
4
, lim x
x
x
x
x
→∞
−
+
−
+
− )
(
= −∞
7
33
51
19
1
9
8
7
4
To determine the end behavior of a polynomial, determine the end behavior of the
highest-powered term.
As x approaches –∞, you have
lim
lim
x
x
x
x
x
x
x
→−
→
∞
− ∞
−
+
−
+
− )
(
=
−
( )
= ∞
7
33
51
19
1
7
9
8
7
4
9
Note that the limit is positive because a negative number raised to an odd power is
negative, but after multiplying by –7, the answer becomes positive.
158
Answers
1–100
80.
domain: −∞ −
(
)∪ ∞
[ )
, 1
2, ; range: − −
(
)∪ [ )
3 1
2 4
,
,
Notice that the function is continuous on its domain.
For the portion of the graph that’s to the left of the y-axis, there’s no largest x value
due to the hole at (–1, –1); instead, the x values get arbitrarily close to –1. The graph
then extends indefinitely to the left so that the domain for the left side of the graph is
(–∞, –1). There’s also no largest y value due to the hole at (–1, –1); the y values get
arbitrarily close to –1. The graph then decreases as it approaches the horizontal
asymptote at y = –3, so the range for this portion of the graph is (–3, –1).
For the portion of the graph that’s to the right of the y-axis, the smallest x value is at
the point (2, 2). The graph then extends indefinitely to the right, so the domain for the
right side of the graph is [2, ∞). The smallest y value also occurs at the point (2, 2), and
then the graph increases as it approaches the horizontal asymptote at y = 4; therefore,
the range of this portion of the graph is [2, 4).
Putting the two parts together, the domain of the function is −∞ −
(
)∪ ∞
[ )
, 1
2, , and the
range is − −
(
)∪ [ )
3 1
2 4
,
, .
81.
lim
x
x
x
→−∞
−
+
(
) = ∞
3
40
33
6
5
, lim x
x
x
→∞
−
+
(
) = ∞
3
40
33
6
5
To determine the end behavior of a polynomial, you just have to determine the end
behavior of the highest-powered term.
As x approaches –∞, you have
lim
lim
x
x
x
x
x
→−
→
∞
− ∞
−
+
(
) = ( )
= ∞
3
40
33
3
6
5
6
Note that even though the limit is approaching negative infinity, the limit is still positive due to the even exponent.
As x approaches ∞, you have
lim
lim x
x
x
x
x
→∞
→∞
−
+
(
) = ( )
= ∞
3
40
33
3
6
5
6
82.
lim
x
x
x
x
x
→−∞
−
+
−
+
− )
(
= ∞
7
33
51
19
1
9
8
7
4
, lim x
x
x
x
x
→∞
−
+
−
+
− )
(
= −∞
7
33
51
19
1
9
8
7
4
To determine the end behavior of a polynomial, determine the end behavior of the
highest-powered term.
As x approaches –∞, you have
lim
lim
x
x
x
x
x
x
x
→−
→
∞
− ∞
−
+
−
+
− )
(
=
−
( )
= ∞
7
33
51
19
1
7
9
8
7
4
9
Note that the limit is positive because a negative number raised to an odd power is
negative, but after multiplying by –7, the answer becomes positive.