First slide

Functions (XII)

Question

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

Moderate

A

even function
B

odd function
C

neither even nor odd
D

periodic function

Solution

$\begin{array}{l} f (x) = \sin (\log (x + \sqrt{1 + x^{2}})) \\ or f (- x) = \sin [\log (- x + \sqrt{1 x^{2}})] \\ = \sin (\log (\sqrt{1 + x^{2}} - x \frac{(\sqrt{1 + x^{2}} + x)}{(\sqrt{1 + x^{2}} + x)})) \\ = \sin (\log [\frac{1}{(x + \sqrt{1 + x^{2}})}]) \end{array}$
$\begin{array}{l} = sinlog [- \log (x + \sqrt{1 + x^{2}})] \\ = - \sin [\log (x + \sqrt{1 + x})] \end{array}$
$∴ f (- x) = - f (x)$
Therefore, f(x) is an odd function.

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