Question

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

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

Answer 1

The correct option is A an even function

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

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

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

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

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

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

$⟹f(-x)=-f\left(x\right)$

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

Answer-(B)