To find the formula for $g(x)$, the function $f(x) = 4\sqrt{x}$ needs to be shifted up 3 units and to the right 3 units.

Step-by-step process:

1. Shift right by 3 units:
   To shift a function horizontally to the right by 3 units, replace $x$ with $x - 3$ in the function.
   So, shifting $f(x) = 4\sqrt{x}$ to the right by 3 units gives:
   $f(x - 3) = 4\sqrt{x - 3}$

2. Shift up by 3 units:
   To shift a function vertically up by 3 units, add 3 to the entire function.
   So, shifting $4\sqrt{x - 3}$ up by 3 units gives:
   $g(x) = 4\sqrt{x - 3} + 3$

Thus, the formula for $g(x)$ is: $g(x) = 4\sqrt{x - 3} + 3$

Would you like further details or have any questions?

Related Questions:

1. What is the graph of $f(x) = 4\sqrt{x}$ before any shifts?
2. How does shifting a function horizontally or vertically affect its domain and range?
3. What is the domain of $g(x) = 4\sqrt{x - 3} + 3$?
4. How would the graph change if we shifted $f(x)$ left by 3 units instead?
5. How do we determine the inverse of the function $f(x)$?

Tip: Always check the domain of a function after performing transformations, as shifts can alter where the function is defined.