We are given the function:

$f(x) = 2\sqrt{x}$

and we need to determine the function $g(x)$ after shifting it up 3 units and left 5 units.

Step 1: Left Shift by 5 Units

Shifting a function left by $h$ units means replacing $x$ with $x + h$ in the function. Since we shift left by 5:

$f(x+5) = 2\sqrt{x+5}$

Step 2: Upward Shift by 3 Units

Shifting up by $k$ units means adding $k$ to the function:

$g(x) = f(x+5) + 3$

Substituting $f(x+5)$:

$g(x) = 2\sqrt{x+5} + 3$

Thus, the formula for $g(x)$ is:

$\boxed{g(x) = 2\sqrt{x+5} + 3}$

Would you like a graph or further explanation?

Related Questions:

1. How would the function change if it were shifted down instead of up?
2. What is the domain of $g(x)$?
3. How do horizontal and vertical shifts affect the range of $f(x)$?
4. What are the new x- and y-intercepts of $g(x)$?
5. If we reflect $g(x)$ across the x-axis, what is the new equation?

Tip: When shifting a function horizontally, always modify the inside of the function. When shifting vertically, modify the outside of the function.