NIMCET 2017 — Mathematics PYQ

$\text{Calculation:}$  

$\text{We have }f(x)=\log \left(x+\sqrt{x^2+1}\right)$  

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

$\text{Now, }f(x)+f(-x)=\log \left(x+\sqrt{x^2+1}\right)+\log \left(-x+\sqrt{x^2+1}\right)$  

$=\log \left[\left(x+\sqrt{x^2+1}\right)\left(-x+\sqrt{x^2+1}\right)\right]$  

$=\log \left[{\left(\sqrt{x^2+1}\right)}^2-x^2\right]$  

$=\log 1=0$  

$\Rightarrow f(-x)=-f(x)$  

$\text{Therefore, }f(x)\text{ is an odd function.}$