The expression in the image you provided is:

$z = \ln\left(R^2 - x^2 - y^2\right)$

To find the domain of this function, we need to ensure that the argument inside the logarithm is positive, because the natural logarithm (ln) is only defined for positive real numbers. Therefore, the condition for the function to be defined is:

$R^2 - x^2 - y^2 > 0$

This simplifies to:

$x^2 + y^2 < R^2$

This inequality describes the region inside a circle with radius $R$ centered at the origin in the $xy$-plane.

Domain:

The domain of the function is the interior of a circle with radius $R$ centered at the origin, or more formally:

$D = \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < R^2\}$

To graph the domain:

The graph of the domain will be a disk with radius $R$ in the $xy$-plane. The boundary of this disk will be the circle $x^2 + y^2 = R^2$, but the domain only includes points strictly inside this circle.

Would you like to proceed with a plot of this region, or is there anything else you need clarification on?

Here are some related questions:

1. How does the natural logarithm function behave near the boundary of the domain?
2. What would happen to the domain if the inequality changed to $x^2 + y^2 \leq R^2$?
3. How does the domain change if we replace $R$ with a variable function of $x$ and $y$?
4. Can this function have an undefined region when considering complex numbers?
5. What is the geometric interpretation of the logarithmic function in this context?

Tip: Always ensure that the argument inside a logarithmic function is strictly positive to avoid domain issues.