2. Given $f(x)=\frac{a}{a^{2}-2} \cdot\left(a^{x}-a^{-x}\right)(a>0$ and $a \neq 1)$ is an increasing function on $\mathbf{R}$. Then the range of the real number $a$ is ( ).
(A) $(0,1)$
(B) $(0,1) \cup(\sqrt{2},+\infty)$
(C) $(\sqrt{2},+\infty)$
(D) $(0,1) \cup[\sqrt{2},+\infty)$
2.B.

Let $x_{1}a^{x_{2}}, a^{-x_{1}}1$ when $a^{x_{1}}0$, then $a>\sqrt{2}$, which meets the conditions.
In summary, the range of values for $a$ is $(0,1) \cup(\sqrt{2},+\infty)$.