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