Mathematics - Functions Question with Solution | TestHub

Let $f\left(x\right)=y$, then

$a^y=x+\sqrt{x^2+1}$

$\Rightarrow a^{-y}=\frac{1}{x+\sqrt{x^2+1}}$

$\Rightarrow a^{-y}=\frac{x-\sqrt{x^2+1}}{-1}$  [rationalising]

$\therefore a^y-a^{-y}=2x$

$\Rightarrow x=\frac{1}{2}\left(a^y-a^{-y}\right)$

$\therefore f^{-1}\left(y\right)=\frac{1}{2}\left(a^y-a^{-y}\right)$   $\left[\because f\left(x\right)=y\Rightarrow x=f^{-1}\left(y\right)\right]$

$\Rightarrow f^{-1}\left(x\right)=\frac{1}{2}\left(a^x-a^{-x}\right)$  [replacing $y$ by $x$]

Hence, option (a) is correct.