3. Determine the parity of the function $f(x)=\log _{a}\left(x+\sqrt{1+x^{2}}\right)$.
Solution: $f(-x)=\log _{a}\left(-x+\sqrt{1+x^{2}}\right)$,
$$
\because f(-x) \neq-f(x), f(-x) \neq f(x) \text {, }
$$
$\therefore f(x)$ has no odd or even properties.
Analysis: This problem makes a hasty judgment that $f(x)$ has no odd or even properties based on the superficial result without necessary equivalent transformation and simplification, which is inappropriate.
In fact, $f(-x)=\log _{a}\left(-x+\sqrt{1+x^{2}}\right)$
$$
\begin{array}{l}
=\log _{a} \frac{1}{x+\sqrt{1+x^{2}}} \\
=-\log _{a}\left(x+\sqrt{1+x^{2}}=-f(x),\right.
\end{array}
$$
$\therefore f(x)$ is an odd function.