[tex]E(x) =\dfrac{2\left\lfloor x \right\rfloor -x}{\left\lfloor x \right\rfloor + |x| + 2}\\ \\ \\\boxed{1}\quad -1\leq x < 0:\\\\ E(x) = \dfrac{2\cdot (-1)-x}{-1-x+2} = \dfrac{-2-x}{1-x} = \dfrac{x+2}{x-1}\\ \\\\ \boxed{2}\quad 0\leq x<1:\\\\ E(x)= \dfrac{2\cdot 0 - x}{0+x+2} = \dfrac{-x}{x+2}\\ \\ \\\boxed{3}\quad 1\leq x< 2:\\ \\ E(x) = \dfrac{2\cdot 1-x}{1+x+2} = \dfrac{2-x}{x+3}\\ \\\\ \boxed{4}\quad x = 2:\\ \\E(x) = \dfrac{2\cdot 2-x}{2+x+2} = \dfrac{4-x}{4+x}[/tex]
[tex]\\[/tex]
[tex]\Rightarrow E(x) =\begin{cases}\dfrac{x+2}{x-1},\quad x\in [-1,0) \\\\\dfrac{-x}{x+2},\quad x\in [0,1) \\ \\\dfrac{2-x}{x+3},\quad x\in [1, 2)\\ \\ \dfrac{1}{3},\quad x = 2\end{cases}[/tex]
