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If f(x)=log(e)((1-x)/(1+x)),|x| lt 1, " ...

If `f(x)=log_(e)((1-x)/(1+x)),|x| lt 1, " then " f((2x)/(1+x^(2)))` is equal to

A

`2f(x)`

B

`2f(x^(2))`

C

`(f(x))^(2)`

D

`-2f(x)`

Text Solution

Verified by Experts

The correct Answer is:
A
Given, `f(x)-log_(e)((1-x)/(1+x)),|x| lt 1,` then
`f((2x)/(1+x^(2)))=log_(e)((1-(2x)/(1+x^(2)))/(1+(2x)/(1+x^(2)))) " "[ because |(2x)/(1+x^(2))| lt 1]`
`=log_(e)(((1+x^(2)-2x)/(1+x^(2)))/((1+x^(2)+2x)/(1+x^(2))))=log_(e)(((1-x)^(2))/((1+x)^(2)))=log_(e)((1-x)/(1+x))^(2)`
`=2 log_(e)((1-x)/(1+x)) " " [ because log_(e)|A|^(m)=m log_(e)|A|]`
`=2f(x) " " [ because f(x)=log_(e)((1-x)/(1+x))]`
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