Example 15 Find the minimum value of $\sqrt{x^{2}+1}+\sqrt{(4-x)^{2}+4}$.

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Solution: Construct right triangles $\triangle P A C$ and $\triangle P B I$ as shown in Figure 1, such that
$$
\begin{array}{l}
A C=1, B D=2, P C \\
=x, C I=4, \text { and }
\end{array}
$$
$P C$ and $P D$ lie on line $l$. Then the problem of finding the minimum value is converted to "finding a point $I$ on line $l$ such that $P A+P B$ is minimized."
Take the symmetric point $A^{\prime}$ of $A$ with respect to $l$. Clearly,
$$
\text { the original expression }=P A+P B \geqslant A^{\prime} B=5 \text {. }
$$

Therefore, the required minimum value is 5.