Example 3 Find the coordinates of the intersection points of the graph of the function $f(x)=\left|x^{2}-x-2\right|$ and the graph of the function $g(x)=2 x$.
Solve the equation $\left|x^{2}-x-2\right|=2 x$ to get
$$
\left\{\begin{array} { l } 
{ x ^ { 2 } - x - 2 \geqslant 0 , } \\
{ x ^ { 2 } - x - 2 = 2 x }
\end{array} \text { or } \left\{\begin{array}{l}
x^{2}-x-2<0, \\
-\left(x^{2}-x-2\right)=2 x,
\end{array}\right.\right.
$$

Solving this, we get

Therefore, the coordinates of the intersection points of the two function graphs are $(1,2)$ and $\left(\frac{3+\sqrt{17}}{2}, 3+\sqrt{17}\right)$.