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Byju's Answer
Standard XII
Mathematics
One - One function
Let f : ℝ→ℝ b...
Question
Let
f
:
R
→
R
be a function defined by
f
(
x
)
=
{
x
2
+
2
m
x
−
1
,
x
≤
0
m
x
−
1
,
x
>
0
.
If
f
is one-one, then
m
can be
A
−
5
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B
5
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C
4
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D
−
4
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Solution
The correct option is
D
−
4
For
f
to be one-one, vertex of
y
=
x
2
+
2
m
x
−
1
should lie to right of or on
y
−
axis.
⇒
−
2
m
2
≥
0
⇒
m
≤
0
⋯
(
1
)
And for line
y
=
m
x
–
1
m
<
0
⋯
(
2
)
From
(
1
)
and
(
2
)
,
m
<
0
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0
Similar questions
Q.
Let
f
be a real valued function, defined on
R
−
{
−
1
,
1
}
and given by
f
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x
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e
∣
∣
∣
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x
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∣
∣
∣
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Then in which of the following intervals, function
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Q.
Let
f
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R
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R
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f
(
x
)
=
(
|
x
−
1
|
+
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4
x
−
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|
)
[
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]
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[
.
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Q.
Let
f
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S
→
S
where
S
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(
0
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∞
)
be a twice differentiable function such that
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(
x
+
1
)
=
x
f
(
x
)
.
If
g
:
S
→
R
be defined as
g
(
x
)
=
log
e
f
(
x
)
,
then the value of
|
g
′′
(
5
)
−
g
′′
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1
)
|
is equal to :
Q.
Let
f
:
[
0
,
1
]
→
[
0
,
1
]
be defined by
f
(
x
)
=
1
−
x
1
+
x
,
0
≤
x
≤
1
and let
g
:
[
0
,
1
]
→
[
0
,
1
]
be defined by
g
(
x
)
=
4
x
(
1
−
x
)
,
0
≤
x
≤
1
. Determine the composition functions fog(1)
Q.
Let
f
be a twice differentiable function defined on
R
such that
f
(
0
)
=
1
,
f
′
(
0
)
=
2
and
f
′
(
x
)
≠
0
for all
x
∈
R
.
If
∣
∣
∣
f
(
x
)
f
′
(
x
)
f
′
(
x
)
f
′′
(
x
)
∣
∣
∣
=
0
,
for all
x
∈
R
,
then the value of
f
(
1
)
lies in the interval
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