If f(x)=
e^x+e^-x/2
then the inverse f(x) is
Answers
Answered by
1
f(x)=
e
x
+e
−x
e
x
−e
−x
+2
Let f(x)=y
Then, y=
e
x
−e
−x
e
x
−e
−x
+2
y=
e
x
−e
−x
e
x
−e
−x
+2e
x
+2e
−x
⇒ye
x
+ye
−x
=3e
x
+e
−x
⇒ye
x
+
e
x
y
=3e
x
+
e
x
1
⇒y.e
2x
+y=3e
2x
+1
⇒y.e
2x
−3e
2x
=1−y
⇒e
2x
(y−3)=1−y
⇒e
2x
=
y−3
1−y
2x=log
e
(
y−3
1−y
)
∴x=
2
1
log
e
(
y−3
1−y
)
y=f(x)⇒x=f
−1
(y)
∴f
−1
(y)=
2
1
log
e
(
y−3
1−y
)
∴f
−1
(x)=
2
1
log
e
(
x−3
1−x
)
=
2
1
log
e
(3−x
x−1)
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