The problem in the image asks for the equation of a function $g(x)$ which is derived from the given function $f(x) = x^2$. The transformations described are:

1. Horizontal compression by a factor of 3: When a function is horizontally compressed by a factor of 3, the transformation involves multiplying the input by 3. The function becomes $f(3x)$.

2. Shift to the right by 4 units: A horizontal shift to the right by 4 units translates the function by subtracting 4 from the input. So the function becomes $f(3(x - 4)) = (3(x - 4))^2$.

Thus, the equation of the new function $g(x)$ after these transformations is:

$g(x) = (3(x - 4))^2$

Would you like further details or have any other questions?

Here are five related questions for further exploration:

1. What would the equation be if the function was shifted left instead of right?
2. How would the graph change if it was reflected over the x-axis?
3. What happens to the vertex of the parabola after the transformations?
4. How does a vertical stretch differ from a horizontal compression?
5. Can you describe the effect of applying both a vertical and horizontal stretch to the graph?

Tip: When dealing with function transformations, always consider the order of operations: perform stretching/compressing first, followed by shifting.