A) neither an even nor an odd function
B) an even function
C) an odd function
D) a periodic function
Correct Answer: C
Solution :
\[f(x)=log\,(x+\sqrt{{{x}^{2}}+1})\] \[f(-x)=log\,\left\{ (-x+\sqrt{{{x}^{2}}+1}) \right\}=log\,\left\{ \frac{-{{x}^{2}}+{{x}^{2}}+1}{x+\sqrt{{{x}^{2}}+1}} \right\}\] \[=\,\,-log\,(x+\sqrt{{{x}^{2}}+1}\,\,\,=-\,f(x)\] \[\Rightarrow \]f(x) is an odd function.
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