Question

If $\tanh (x)=\frac{1}{3}$ then $\tanh (3 x)$ is

• A. $\frac{8}{9}$
• B. $\frac{7}{9}$
• C. $1$
• D. $\frac{2}{3}$

Answer 1

Answer: (B)

$\tanh (x)=\frac{1}{3}, \tanh (3 x)=?$
$\Rightarrow \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}=\frac{1}{3}$
$\Rightarrow 2 e^{x}=4 e^{-x}$
$\Rightarrow \left(e^{2 x}=2\right)$ and
$\tanh (3 x)=\frac{e^{3 x}-e^{-3 x}}{e^{3 x}+e^{-3 x}}=\frac{\left(e^{2 x}\right)^{3}-1}{\left(e^{2 x}\right)^{3}+1}$
$=\frac{8-1}{8+1}=\frac{7}{9}$