Find polynomials  f(x), g(x) and  h(x) such that for all  x ,  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  F(x) = -1 for all  x<-1 . Also do we know that these functions are unique? How do we know that such functions exist?