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. Odd function
• D. Periodic function

Answer 1

The correct option is B

Odd function

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

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

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

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

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

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

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

$\therefore f\left(x\right)isoddfunction$