15. 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 $a>0$, when 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 $(a, b)$ lies with $a$ and $b$ as coordinates.
15. (1) \( g(x) = \begin{cases} 
0, & x \leqslant -2 \text{ or } x \geqslant 1, \\ 
-x^2 - x + 2, & -2 < x < 1 
\end{cases} \)

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:
\[
\begin{cases}
\varphi(-2) > 0, \\
\varphi\left(-\frac{1}{2}\right) > 0, \\
\varphi(1) > 0,
\end{cases}
\]
which is equivalent to:
\[
\begin{cases}
b > 2a, \\
b > -a.
\end{cases}
\]
Given \( a > 0 \),

Thus, \( 2a < b < \frac{1}{4}(a+1)^2 + 2 \) and \( 0 < a < 3 \) are the required conditions.
The region where the point \((a, b)\) lies, with \(a\) and \(b\) as coordinates, is the shaded area in Figure 1 (excluding the boundaries).