C01S03.017:
The domain of
f
(
x
) =
x
√
x
+ 2 is the interval [
−
2
,
+
∞
), so its graph must be the one
shown in Fig. 1.3.38.
C01S03.018:
The domain of
f
(
x
) =
√
2
x
−
x
2
consists of those numbers for which 2
x
−
x
2
0; that is,
x
(2
−
x
)
0. This occurs when
x
and 2
−
x
have the same sign and also when either is zero. If
x >
0 and
2
−
x >
0, then 0
< x <
2. If
x <
0 and 2
−
x <
0, then
x <
0 and
x >
2, which is impossible. Hence the
domain of
f
is the closed interval [0
,
2]. So the graph of
f
must be the one shown in Fig. 1.3.36.
C01S03.019:
The domain of
f
(
x
) =
√
x
2
−
2
x
consists of those numbers
x
for which
x
2
−
2
x
0; that
is,
x
(
x
−
2)
0. This occurs when
x
and
x
−
2 have the same sign and also when either is zero. If
x >
0
and
x
−
2
>
0, then
x >
2; if
x <
0 and
x
−
2
<
0, then
x <
0. So the domain of
f
is the union of the two
intervals (
−∞
,
0] and [2
,
+
∞
). So the graph of
f
must be the one shown in Fig. 1.3.39.
C01S03.020:
The domain of
f
(
x
) = 2(
x
2
−
2
x
)
1
/
3
is the set
R
of all real numbers because every real
number has a [unique] cube root. By the analysis in the solution of Problem 19,
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 Spring '11
 CHENG
 Calculus, Topology, Complex number, Good viewing window

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