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Byju's Answer
Standard XII
Mathematics
Definition of Function
Show that the...
Question
Show that the function
f
:
R
→
R
defines by
f
(
x
)
=
x
x
2
+
1
,
∀
x
∈
R
, is neither one-one nor onto.
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Solution
For
x
1
,
x
2
∈
R
such that
x
1
≠
x
2
−
1
Consider
f
(
x
1
)
=
f
(
x
2
)
⇒
x
1
x
2
1
+
1
=
x
2
x
2
2
+
1
⇒
x
1
x
2
2
+
x
1
=
x
2
x
2
1
+
x
2
⇒
x
1
x
2
(
x
2
−
x
1
)
−
(
x
2
−
x
1
)
=
0
⇒
(
x
1
x
2
−
1
)
(
x
2
−
x
1
)
=
0
⇒
x
1
x
2
=
1
⇒
x
1
=
1
x
2
Hence,
f
is not one-one as
∀
x
1
∈
R
there exists
x
2
such that
f
(
x
1
)
=
f
(
x
2
)
Let,
f
(
x
)
=
y
⇒
x
x
2
+
1
=
y
⇒
y
(
x
2
+
1
)
=
x
⇒
y
x
2
−
x
+
1
=
0
⇒
x
=
1
±
√
1
−
d
y
2
2
y
Since,
x
∈
R
∴
1
−
4
y
2
≥
0
⇒
(
1
−
2
y
)
(
1
−
2
y
)
≥
0
⇒
−
1
2
≤
y
≤
1
2
So,
Range
(
f
)
∈
[
−
1
2
,
1
2
]
≠
R
Hence,
f
is not into.
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Definition of Function
Standard XII Mathematics
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