55.
[
2
5, 5] by [
2
2, 5]
We require
x
2
2
4
$
0 (so that the square root is defined)
and
x
2
2
4
±
0 (to avoid division by zero), so the domain
is (
2‘
,
2
2)
<
(2,
‘
). For values of
x
in the domain,
x
2
2
4
1
and hence
ˇ
x
2
w
2
w
4
w
and
}
ˇ
x
2
w
1
2
w
4
w
}
2
can attain any positive
value, so the range is (0,
‘
). (Note that grapher failure may
cause the range to appear as a finite interval on a
grapher.
56.
[
2
5, 5] by [
2
2, 5]
We require 9
2
x
2
$
0 (so that the fourth root is defined)
and 9
2
x
2
±
0 (to avoid division by zero), so the domain
is (
2
3, 3). For values of
x
in the domain, 9
2
x
2
can attain
any value in (0, 9]. Therefore,
ˇ
4
9
w
2
w
x
w
2
w
can attain any
value in (0,
ˇ
3
w
], and
}
ˇ
4
9
w
2
2
w
x
w
2
w
}
can attain any value in
3
}
ˇ
2
3
w
}
,
‘
2
. The range is
3
}
ˇ
2
3
w
}
,
‘
2
or approximately [1.15,
‘
).
(Note that grapher failure may cause the range to appear as
a finite interval on a grapher.)
57.
[
2
4.7, 4.7] by [
2
3.1, 3.1]
We require 9
2
x
2
±
0, so the domain is
(
2‘
,
2
3)
<
(
2
3, 3)
<
(3,
‘
). For values of
x
in the
domain, 9
2
x
2
can attain any value in
(
2‘
, 0)
<
(0, 9], so
ˇ
3
9
w
2
w
x
w
2
w
can attain any value in
(
2‘
, 0)
<
(0,
ˇ
3
9
w
]. Therefore,
}
ˇ
3
9
w
2
2
w
x
w
2
w
}
can attain any
value in (
2‘
, 0)
<
3
}
ˇ
3
2
9
w
}
,
‘
2
. The range is
(
2‘
, 0)
<
3
}
ˇ
3
2
9
w
}
,
‘
2
or approximately (
2‘
, 0)
<
[0.96,
‘
).
(Note that grapher failure can cause the intervals in the
range to appear as finite intervals on a grapher.)
58.
[
2
2.35, 2.35] by [
2
1.55, 1.55]
We require
x
2
2
1
±
0, so the domain is
(
2‘
,
2
1)
<
(
2
1, 1)
<
(1,
‘
). For values of
x
in the
domain,
x
2
2
1 can attain any value in [
2
1, 0)
<
(0,
‘
), so
ˇ
3
x
2
w
2
w
1
w
can also attain any value in [
2
1, 0)
<
(0,
‘
).
Therefore,
}
ˇ
3
x
1
w
2
w
1
w
}
can attain any value in
(
2‘
,
2
1]
<
(0,
‘
). The range is (
2‘
,
2
1]
<
(0,
‘
).
(Note that grapher failure can cause the intervals in the
range to appear as finite intervals on a grapher.)
59. (a)
(b)
60. (a)
y
x
2
2
1
–1
–1
–2
–2
y
x
1.5
2
0
–2
–1.5
y
x
1.5
2
0
–2
–1.5
Section 1.2
11