Question

$f:R\to R$ such that $f\left(x\right)=\ell n(x+\sqrt{x^2+1})$. Another function $g\left(x\right)$ is defined such that $gof\left(x\right)=x\forall x\in R$. Then $g\left(2\right)$ is -

• A. $\frac{e^2+e^{-2}}{2}$
• B. $e^2$
• C. $\frac{e^2-e^{-2}}{2}$
• D. $e^{-2}$

Answer 1

The correct option is C $\frac{e^2-e^{-2}}{2}$

$f\left(x\right)=ln(x+\sqrt{x^2+1})$

$g\left(x\right)=g\left(f\right(x\left)\right)$ $g\left(2\right)=?$

ln $(x+\sqrt{x^2+1})=y$

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

$x^2+1=e^{2y}-2e^{y}x+x^2$

$e^{2y}-2e^{y}x-1=o\Rightarrow$

$x=\frac{e^{2y}-1}{2e^y}$

$=\frac{(e^y-e^{-y})}{2}$

$\therefore g\left(x\right)=\frac{e{}^y-e^{-y}}{2}\Rightarrow g\left(2\right)=\frac{e^2-e^{-2}}{2}$