11. (20 points) Given the function $f(x)=x^{2}+x-2$.
(1) Try to find the analytical expression of the function $g(x)=\frac{|f(x)|-f(x)}{2}$;
(2) If when $a>0$, the line $y=a x+b$ intersects the curve $y=g(x)$ at three different points, try to determine the relationship between $a$ and $b$, and draw a diagram to represent the region where the point $(a, b)$ with coordinates $a$ and $b$ lies.
11. (1)
$$
g(x)=\left\{\begin{array}{ll}
0, & x \leqslant-2 \text { or } x \geqslant 1, \\
-x^{2}-x+2, & -2 < x < 1.
\end{array}\right.
$$

When $a > 0$, the line $y = ax + b$ intersects the curve $y = g(x)$ at three different points if and only if the line intersects the curve in the interval $-2 < x < 1$. This requires:
$$
\left\{\begin{array}{l}
\varphi(-2) > 0, \\
\varphi\left(-\frac{1}{2}\right) < 0, \\
\varphi(1) > 0
\end{array} \Rightarrow \left\{\begin{array}{l}
b > 2a, \\
b < \frac{1}{4}(a+1)^{2} + 2, \\
b > -a.
\end{array}\right.\right.
$$

Given $a > 0$, we have $2a < b < \frac{1}{4}(a+1)^{2} + 2$, and $0 < a < 3$. The region where the point $(a, b)$ lies is the shaded area in Figure 5 (excluding the boundaries).