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