Question

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

• A. an even function
• B. an odd function
• C. a periodic function
• D. none of these

Answer 1

The correct option is B an odd function

$f\left(x\right)=\log (x+\sqrt{x^2+1^2})f(-x)=\log (-x+\sqrt{1+x^2})=\log \left(\frac{(\sqrt{1^2+x^2}-x)(\sqrt{1+x^2}+x)}{\sqrt{1+x^2}+x}\right)f(-x)=\log \left(\frac{1}{x+\sqrt{1+x^2}}\right)f(-x)=-\log (x+\sqrt{1+x^2})\because f(-x)=-f\left(x\right)\therefore functionisoddfunction.$