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Question

Let f:RR be defined by f(x)=(exex)2.
The inverse of the given function is:

A
f1(x)=loge(x+x2+1)
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B
f1(x)=loge(xx2+1)
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C
f1(x)=loge(x+x21)
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D
f1(x)=loge(xx21)
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Solution

The correct option is A f1(x)=loge(x+x2+1)
Let y=f(x)=(exex)/2
2y=exex
e2x2yex1=0
ex=2y±4y2+42=y±y2+1
ex=y+y2+1
(yy2+1<0 for all y but ex is always positive)
x=loge(y+y2+1)
f1(x)=loge(x+x2+1)

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