Question

Which of the following statements is/are true?

$I$: Every function must be either even or odd function
$II$: The function $f\left(x\right)=\log (x+\sqrt{x^2+1})$ is an odd function.

• A. Only $I$ is true
• B. Only $II$ is true
• C. Both $I$ and $II$ are true
• D. Neither $I$ nor $II$ is true

Answer 1

The correct option is B Only $II$ is true

$I$
A function can be odd, even or neither odd nor even.
$II$
$f\left(x\right)=\log (x+\sqrt{x^2+1})$
$f(-x)=\log (\sqrt{x^2+1}-x)$
$f(-x)=\log \left(\right(\sqrt{x^2+1}-x)\times \frac{\sqrt{x^2+1}+x}{\sqrt{x^2+1}+x})$
$=\log \left(\frac{1}{\sqrt{x^2+1}+x}\right)$
$=-\log (x+\sqrt{x^2+1})$
$f(-x)=-f\left(x\right)$
$\therefore f\left(x\right)$ is an odd function