Example 12 Find the range of the function $y=3 x+2 \sqrt{x^{2}-2}$.
Let $u=3 x, v=2 \sqrt{x^{2}-2}(v \geqslant 0,|u| \geqslant 3 \sqrt{2})$, then $y=u+v$ and $\frac{u^{2}}{18}-\frac{v^{2}}{8}=1$.
Considering $y$ as a parameter, in the $u O v$ coordinate system, draw the line system $v=-u+y$ and the part of the hyperbola $\frac{u^{2}}{18}-\frac{v^{2}}{8}=1$ $(v>0)$, as shown in Figure 29-3.

When the line passes through the point $(3 \sqrt{2}, 0)$, the intercept $y$ on the $v$-axis is $y=3 \sqrt{2}$; when the line is tangent to the left branch of the hyperbola, by the tangent formula $y=k x \pm \sqrt{k^{2} a^{2}-b^{2}}$, the intercept $y$ is $y=-\sqrt{10}$.
Therefore, the range of the function $y$ is $(-\infty,-\sqrt{10}) \cup[3 \sqrt{2},+\infty)$.