Answer
See graph (red). domain $(-\infty,0)\cup (0,\infty)$, range $(-\infty,0)\cup (0,\infty)$.
Work Step by Step
Step 1. To obtain the graph of $g(x)=\frac{1}{-x}$ from $y=\frac{1}{x}$, reflect the curve across the y-axis. See graph (red).
Step 2. We can find the domain $(-\infty,0)\cup (0,\infty)$ and the range $(-\infty,0)\cup (0,\infty)$.