Answer
See graph (red).
domain $(-\infty,\infty)$, range $(-\infty,\infty)$.
Work Step by Step
Step 1. To obtain the graph of $f(x)=(x+2)^3-3$ from $y=x^3$, shift the curve 2 units to the left, then 3 units down. See graph (red).
Step 2. We can find the domain $(-\infty,\infty)$ and the range $(-\infty,\infty)$.