Example 2. Given: $f(x)=2 x+3,|x| \leqslant 2$, and $x \in Z$. Find the range of the function.
$$
\begin{array}{c}
\text { Solve } \because|x| \leqslant 2, \text { and } x \in Z, \\
\therefore x \text { can only be }-2,-1,0,1,2 . \\
\text { Also } \because f(-2)=-1, f(-1)=1, f(0) \\
=3, f(1)=5, f(2)=7,
\end{array}
$$
$\therefore$ the range of the function is $\{-1,1,3,5,7\}$.
$$