Question

Find inverse $f\left(x\right)={\log }_e(x+\sqrt{x^2+1})$

• A. $\sinh \left(x\right)$
• B. $\cosh \left(x\right)$
• C. $\tanh \left(x\right)$
• D. $\coth \left(x\right)$

Answer 1

Answer: (A)

$\sinh \left(x\right)$

$y=\log (x+\sqrt{x^2+1})$
$\Rightarrow e^y=x+\sqrt{x^2+1}$
$\Rightarrow e^y-x=\sqrt{x^2+1}$
$\Rightarrow x^2-2x{e}^y+e^{2y}=x^2+1$
$\Rightarrow e^{2y}=1+2x{e}^y$
$\Rightarrow 2x=\frac{e^{2y}-1}{e^y}$
$\Rightarrow x=\frac{e^y-e^{-y}}{2}$
Replacing x with y
$f^{-1}\left(x\right)=\frac{e^x-e^{-x}}{2}$
$=\sinh \left(x\right)$