Question

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

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

Answer 1

Answer: (A)

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

Thus $f\left(x\right)$ is odd function