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Byju's Answer
Standard XII
Mathematics
Area between Two Curves
For the funct...
Question
For the function
f
:
R
→
R
,
f
(
x
)
=
3
x
+
2
, if
f
−
1
, exists then find
f
−
1
(
x
)
.
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Solution
Inverse of a function exists if it is both one-one and onto.
Checking for one-one:
if
f
(
x
1
)
=
f
(
x
2
)
⟹
3
x
1
+
2
=
3
x
2
+
2
⟹
x
1
=
x
2
Hence ,
f
(
x
)
is one-one
Co-domain of
f
(
x
)
is
R
And, we know that
3
x
+
2
is a straight line graph and hence its range is
R
Since, range
=
co-domain , hence
f
(
x
)
is onto as well.
Hence,
f
(
x
)
is invertible.
Now,
y
=
f
(
x
)
=
3
x
+
2
⟹
x
=
y
−
2
3
And,
x
=
f
−
1
(
y
)
⟹
f
−
1
(
y
)
=
y
−
2
3
Replacing
y
with
x
-
⟹
f
−
1
(
x
)
=
x
−
2
3
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