Question

The inverse of $f \left(x\right)= \frac{2}{3} \frac{10^{x}-10^{-x}}{10^{x}+ 10^{-x}}$ is

• A. $\frac{1}{3}\,log_{10} \frac{1+x}{1-x}$
• B. $\frac{1}{2}\,log_{10} \frac{2+3x}{2-3x}$
• C. $\frac{1}{3}\,log_{10} \frac{2+3x}{2-3x}$
• D. $\frac{1}{6}\,log_{10} \frac{2-3x}{2+3x}$

Answer 1

Answer: (B)

If $y = \frac{2}{3} \frac{10^{x}-10^{-x}}{10^{x}+ 10^{-x}}, \quad10^{2x} = \frac{3y+2}{2-y}$
or $x = \frac{1}{2}\,log_{10} \frac{2+3y}{2-3y} \therefore f^{-1} \left(x\right) = \frac{1}{2}\,log_{10} \frac{2+3x}{2-3x}.$