Example 12. Let
$$
\begin{array}{l}
\boldsymbol{f}(x)=\left[\begin{array}{c}
x \\
12 \frac{1}{2}
\end{array}\right] \cdot\left[\frac{-12 \frac{1}{2}}{x}\right] \text {, } \\
x \in(0,90) . \\
\end{array}
$$

Find the range of $f(x)$.
When $x<12 \frac{1}{2}$, $0<\frac{x}{12}<1$, then $f(x)=0$.
When $x \geqslant 12 \frac{1}{2}$, $-1 \leqslant \frac{-12 \frac{1}{2}}{x}<0$, then
$$
\begin{array}{l}
{\left[\begin{array}{c}
-12 \frac{1}{2} \\
x
\end{array}\right]=-1} \\
f(x)=-\left[\frac{x}{12 \frac{1}{2}}\right] . \\
\end{array}
$$

By $12 \frac{1}{2} \leqslant x<90$, then
$$
f(x) \text { can get }-1,-2,-3,-4,-5,-6 \text {, }
$$
-7. That is, the range of $f(x)$ is
$$
\{0,-1,-2,-3,-4,-5,-6,-7\}
$$
(Author: Tianjin Beimen Dong Middle School)