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