11. Let $x>0$, then the range of the function $f(x)=\frac{x+x^{-1}}{[x] \cdot\left[x^{-1}\right]+[x]+\left[x^{-1}\right]+1}$ is $\qquad$
11. $\left\{\frac{1}{2}\right\} \cup\left[\frac{5}{6}, \frac{5}{4}\right)$ By symmetry, we can assume $x \geqslant 1$.
（1）When $x=1$, $f(x)=\frac{1}{2}$;
（2）When $x>1$, let $x=n+\alpha$, where $n \in \mathbf{N}_{+}, 0 \leqslant \alpha<1$. Since $f(x)$ is an increasing function when $x>1$.
Considering function (1), since $n \leqslant n+\alpha$, and $a_{2}=a_{3}, a_{3}<a_{4}<\cdots$, we have $b_{1}>b_{2}>b_{3}>b_{4}>\cdots$, which means $b_{1}$ is the largest.
Therefore, when $x>1$, the range of $f(x)$ is $\left[a_{2}, b_{1}\right)=\left[\frac{5}{6}, \frac{5}{4}\right)$.
Thus, the range of $f(x)$ is $\left\{\frac{1}{2}\right\} \cup\left[\frac{5}{6}, \frac{5}{4}\right)$.