The inverse of the function `f: Rvec{x in R : x<1}` given by `f(x)=(e^x-e^(-x))/(e^x+e^(-x)),` is `1/2log(1+x)/(1-x)` (b) `1/2log(2+x)/(2-x)` `1/2log(1-x)/(1+x)` (d) None of these

Explanation

`f(x)=frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}`


Let `f^{-1}(x)=y quad cdots(1)`


`Rightarrow {f}({y})={x}`


`Rightarrow frac{e^{y}-e^{-}}{e^{y}+e^{-}}=x`


`Rightarrow frac{e^{-y}(e^{2 y}-1)}{e^{-y}(e^{2 y}+1)}=x`


`Rightarrow(e^{2} y-1)=x(e^{2 y}+1)`


`Rightarrow {e}^{2 y}-1=x {e}^{2 {y}}+{x}`


`Rightarrow {e}^{hat{y}}=frac{1+{x}}{1-{x}}`


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