Question

The inverse of the function is $f\left(x\right)=\frac{{10}^x-{10}^{-x}}{{10}^x+{10}^{-x}}+1$ is

• A. $f^{-1}\left(x\right)=lo{g}_{10}\frac{x}{2-x}$
• B. $f^{-1}\left(x\right)=\frac{1}{2}lo{g}_{10}\frac{x}{2-x}$
• C. $f^{-1}\left(x\right)=\frac{1}{2}lo{g}_{10}\frac{x}{1-x}$
• D. $Noneofthese$

Answer 1

The correct option is B $f^{-1}\left(x\right)=\frac{1}{2}lo{g}_{10}\frac{x}{2-x}$

Let f(x) = y
$\Rightarrow \frac{{10}^x-{10}^{-x}}{{10}^x+{10}^{-x}}+1=y$
Applying componendo and divendo -
$\Rightarrow x=\frac{1}{2}lo{g}_{10}\frac{y}{2-y}\Rightarrow f^{-1}\left(x\right)=\frac{1}{2}lo{g}_{10}\frac{x}{2-x}$