Answer
See graph (red).
domain $(-\infty,\infty)$, range $[-3,\infty)$.
Work Step by Step
Step 1. To obtain the graph of $g(x)=3|x+1|-3$ from $y=|x|$, shift the curve 1 unit to the leftt, stretch vertically by a factor of 3, then shift 3 unit down. See graph (red).
Step 2. We can find the domain $(-\infty,\infty)$ and the range $[-3,\infty)$.