Example 9 Find the minimum value of the function $y=\sqrt{2 x^{2}-3 x+1}+$ $\sqrt{x^{2}-2 x}$.
Solution: First, find the domain $(-\infty, 0] \cup[2,+\infty)$, note that the functions inside the two square roots are both decreasing on $(-\infty, 0]$ and increasing on $[2,+\infty)$, so the original function is also such. Therefore, $y_{\text {min }}$ $=\min \{f(0), f(2)\}=1$. The minimum value is achieved when $x=0$.