Find polynomials $\displaystyle f(x), g(x) $ and $\displaystyle h(x) $ such that for all $\displaystyle x $, $\displaystyle F(x) = |f(x)|-|g(x)|+h(x) = \begin{cases} -1 \ \ \ \ \ \ \ \ \text{if} \ x < -1 \\ 3x+2 \ \ \ \ \text{if} \ -1 \leq x \leq 0 \\ -2x+2 \ \ \text{if} \ x>0 \end{cases} $

So it seems that the functions should have a simple form since $\displaystyle F(x) = -1 $ for all $\displaystyle x<-1 $. Also do we know that these functions are unique? How do we know that such functions exist?