Here's the graph of the function $F(x, y)$. It shows two hemispherical shapes:

• The negative hemisphere is centered at $(3, 3)$, and the function takes values from $-4$ to $0$.
• The positive hemisphere is centered at $(-3, -3)$, and the function takes values from $0$ to $4$.

The plot clearly illustrates how the function is defined over two distinct circular regions in the $xy$-plane, with values in the range $[-4, 4]$.

Would you like further details or have any questions? Here are some related topics you might be interested in:

1. How would the graph look if the hemispheres had different radii?
2. What is the effect of shifting the centers of the circles in $F(x, y)$?
3. How do you compute the gradient of $F(x, y)$ at different points?
4. Can we extend $F(x, y)$ to three dimensions by introducing a new variable?
5. How would the range change if we modified the square root terms in $F(x, y)$?

Tip: When graphing functions of two variables, always check the domain carefully to avoid undefined regions, which can lead to computational errors (as seen with the square root).