We can obtain the given function $g(x) = \sqrt{x-2} + 1$ from the simpler function $f(x) = \sqrt{x}$ in two steps:
1. Replace $x$ by $(x-2)$, to obtain $\sqrt{x-2}$. This shifts the graph right 2 units.
2. Add 1 to the function, to obtain the desired function $g(x)$. This shifts the graph up 1 unit.
Thus, the graph of $g(x)$ is obtained from that of $f(x) = \sqrt{x}$ by shifting it right 2 units and up one unit.