find the range of the function 1/4+2sin5x
Answers
Answer:
Note that the denominator is undefined whenever
4
sin
(
x
)
+
2
=
0
,
that is, whenever
x
=
x
1
,
n
=
π
6
+
n
2
π
or
x
=
x
2
,
n
=
5
π
6
+
n
2
π
,
where
n
∈
Z
(
n
is an integer).
As
x
approaches
x
1
,
n
from below,
f
(
x
)
approaches
−
∞
, while if
x
approaches
x
1
,
n
from above then
f
(
x
)
approaches
+
∞
. This is due to division by "almost
−
0
or
+
0
".
For
x
2
,
n
the situation is reversed. As
x
approaches
x
2
,
n
from below,
f
(
x
)
approaches
+
∞
, while if
x
approaches
x
2
,
n
from above then
f
(
x
)
approaches
−
∞
.
We get a sequence of intervals in which
f
(
x
)
is continuous, as can be seen in the plot. Consider first the "bowls" (at whose ends the function blows up to
+
∞
). If we can find the local minima in these intervals, then we know that
f
(
x
)
assumes all the values between this value and
+
∞
. We can do the same for "upside-down bowls", or "caps".
We note that the smallest positive value is obtained whenever the denominator in
f
(
x
)
is as large as possible, that is when
sin
(
x
)
=
1
. So we conclude that the smallest positive value of
f
(
x
)
is
1
4
⋅
1
+
2
=
1
6
.
The largest negative value is similarly found to be
1
4
⋅
(
−
1
)
+
2
=
−
1
2
.
Due to the continuity of
f
(
x
)
in the intervals between discontinuities, and the Intermediate value theorem , we can conclude that the range of
f
(
x
)
is
R
=
(
−
∞
,
−
1
2
]
∪
[
1
6
,
+
∞
)
The hard brackets mean that the number is included in the interval (e.g.
−
1
2
), while soft brackets means that the number is not included.
graph{1/(4sin(x) + 2) [-10, 10, -5, 5]}
Explanation:
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Answer:
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