Express the function $\small{f(x) = 2\!\left|x-3\right|-\left|x+1\right|}$ in piecewise form without using absolute values. [Suggestion: It may help to generate the graph of the function.]

$$\small{f(x) =} \begin{cases} f_1 & x \le a \\ f_2 & a < x < b \\ f_3 & x \ge b \end{cases}$$