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Byju's Answer
Standard XII
Mathematics
Definition of Functions
The function ...
Question
The function
f
:
R
→
[
−
1
2
,
1
2
]
defined as
f
(
x
)
=
x
1
+
x
2
,
is :
A
injective but not surjective.
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B
surjective but not injective.
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C
neither injective nor surjective.
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D
invertible
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Solution
The correct option is
B
surjective but not injective.
f
(
x
)
=
x
1
+
x
2
for checking whether function is injective
f
′
(
x
)
=
(
1
+
x
2
)
(
1
)
−
(
x
)
(
0
+
2
x
)
(
1
+
x
2
)
2
f
′
(
x
)
=
x
2
+
1
−
2
x
2
(
1
+
x
2
)
2
f
′
(
x
)
=
1
−
x
2
1
+
x
2
We Know that , for a function to be one-one its derivative has to always on same side of zero i.e always positive or always negative
f
′
(
x
)
=
1
−
x
2
1
+
x
2
so it is neither always +ve
,
nor always - ve.
∴
f
(
x
)
is not injective.
→
for checking whether function is surjective.
y
=
x
1
+
x
2
⇒
x
2
y
+
y
−
x
=
0
x
=
+
1
±
√
1
−
4
y
2
2
y
we need to make
D
⩾
0
1
−
4
y
2
⩾
0
⇒
(
1
−
2
y
)
(
1
+
2
y
)
⩾
0
⇒
(
2
y
−
1
)
(
2
y
+
1
)
⩽
0
y
∈
[
−
1
2
1
2
]
Range of
f
(
x
)
is same as codomain, So Surjective.
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Similar questions
Q.
The function
f
:
R
→
[
−
1
2
,
1
2
]
defined as
f
(
x
)
=
x
1
+
x
2
,
is
Q.
The function
f
:
R
→
R
,
f
x
=
x
2
is
(a) injective but not surjective
(b) surjective but not injective
(c) injective as well as surjective
(d) neither injective nor surjective
Q.
The function :
R
→
[
−
1
2
,
1
2
]
defined as
f
(
x
)
=
x
1
+
x
2
is
Q.
Let
f
:
R
→
R
be a function defined by
f
(
x
)
=
x
3
+
x
2
+
3
x
+
sin
x
.
Then
f
is
Q.
If
f
:
N
→
N
is a function defined as
f
(
x
)
=
3
x
2
+
2.
Then
f
(
x
)
is
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