Question

The function $f\left(x\right)=\sin \left[\log (x+\sqrt{x^2+1})\right]$ is

• A. even
• B. odd
• C. neither even nor odd
• D. periodic

Answer 1

The correct option is B odd

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

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

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

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

$\Rightarrow f(-x)=\sin [-\log (x+\sqrt{1+x^2})]$

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

$\Rightarrow f(-x)=-f\left(x\right)$

$\Rightarrow f\left(x\right)$ is an odd function.