1. Given $a$ is a positive real number, and $f(x)=\frac{1}{a}-\frac{1}{a^{x}+1}$ is an odd function. Then the range of $f(x)$ is . $\qquad$
$$
\text { - 1. }\left(-\frac{1}{2}, \frac{1}{2}\right) \text {. }
$$

From the fact that $f(x)$ is an odd function, we know
$$
\begin{array}{l}
\frac{1}{a}-\frac{1}{a^{x}+1}=-\frac{1}{a}+\frac{1}{a^{-x}+1} \Rightarrow a=2 \\
\Rightarrow f(x)=\frac{1}{2}-\frac{1}{2^{x}+1}
\end{array}
$$
$\Rightarrow$ The range of $f(x)$ is $\left(-\frac{1}{2}, \frac{1}{2}\right)$.