Determine the domain, range, and graph the function:

[tex]\[ f(x) = \left\{
\begin{array}{clc}
(x+2)^2 + 1 & ; \text{ if } & x \in (-4, 0] \\
\sqrt{x+1} & ; \text{ if } & x \in [0, 4] \\
2x - 8 & ; \text{ if } & x \in [4, \infty)
\end{array}
\right. \][/tex]