To shift the graph of the function $f(x) = 2^x$ to the right by 4 units and up by 3 units, we apply the following transformations:

1. Shift to the right by 4 units: Replace $x$ with $(x - 4)$. This shifts the graph to the right by 4 units.

   The new function becomes: $f(x - 4) = 2^{x - 4}$.

2. Shift up by 3 units: Add 3 to the function to shift it upwards.

   The new function becomes: $f(x - 4) + 3 = 2^{x - 4} + 3$.

Thus, the equation of the function $g(x)$ that shifts the graph of $f(x) = 2^x$ right by 4 units and up by 3 units is: $g(x) = 2^{x - 4} + 3$

Would you like more details on how these transformations work? Here are 5 related questions to deepen your understanding:

1. What happens if we shift the graph left instead of right?
2. How does the base of the exponential function affect transformations?
3. Can you apply similar transformations to other types of functions, like logarithmic or trigonometric functions?
4. What would the equation look like if the graph was shifted down instead of up?
5. How do horizontal stretches or compressions differ from horizontal shifts?

Tip: Shifting a graph left or right involves adjusting the value of $x$ inside the function, while shifting it up or down is done by adding or subtracting outside the function.