The function $f(x) = 2\sqrt{x}$ is shifted down by 4 units and to the right by 2 units to create the new function $g(x)$.

1. Right Shift by 2 Units: To shift a function $f(x)$ to the right by 2 units, we replace $x$ with $x - 2$. This gives us: $f(x - 2) = 2\sqrt{x - 2}$

2. Downward Shift by 4 Units: To shift a function down by 4 units, we subtract 4 from the function. Applying this to $2\sqrt{x - 2}$, we get: $g(x) = 2\sqrt{x - 2} - 4$

Thus, the formula for $g(x)$ is: $g(x) = 2\sqrt{x - 2} - 4$

Would you like more details or have any questions about this? Here are some related questions to consider:

1. What happens if we shift $f(x)$ left or up instead?
2. How would the graph of $g(x)$ look compared to $f(x)$?
3. Can the domain of $g(x)$ be affected by these shifts?
4. How can we generalize the formula for shifting any function horizontally and vertically?
5. What effect would shifting $f(x)$ left by 3 units and up by 5 units have?

Tip: When shifting graphs horizontally, always adjust the variable inside the function. When shifting vertically, adjust the function itself.