Question

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

• A. Even function
• B. Odd function
• C. Neither even nor odd
• D. Periodic function

Answer 1

The correct option is B Odd function

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

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

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

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

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


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


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

$\therefore$ f(x) is odd function.