Question

The inverse of the function $f(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}+2$ is

• A. ${\log }_e{\left(\frac{x-3}{x-1}\right)}^{1/2}$
• B. ${\log }_e{\left(\frac{x-1}{3-x}\right)}^{1/2}$
• C. ${\log }_e{\left(\frac{x+2}{x-3}\right)}^{1/2}$
• D. ${\log }_e{\left(\frac{x+1}{x-2}\right)}^{1/2}$

Answer 1

Answer: (B)

$\frac{y-2}{1}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$
Applying comp. and dividendo:
$\frac{y-1}{3-y}=\frac{2e^x}{2e^{-x}}=e^{2x}$
$\therefore x=\frac{1}{2}\log \left(\frac{y-1}{3-y}\right)=\log {\left(\frac{y-1}{3-y}\right)}^{1/2}$
Hence, the inverse of the function
$f(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}+2$ is
${\log }_c{\left(\frac{x-1}{3-x}\right)}^{1/2}$