To determine the domain of the function $g(x) = (x - 3)^3 - 2$, we need to identify any restrictions on the input values $x$.

Analysis:

The function involves:

1. $(x - 3)^3$ — This is a cubic expression, which is defined for all real values of $x$.
2. Subtracting 2 from the result does not impose any additional restrictions.

Conclusion:

There are no restrictions on the values of $x$. Cubic functions and constants are continuous and defined for all real numbers.

Thus, the domain of $g(x) = (x - 3)^3 - 2$ is: $\boxed{(-\infty, \infty)}$

Would you like more details, or do you have any questions?

Here are 5 related questions for deeper exploration:

1. What is the range of $g(x) = (x - 3)^3 - 2$?
2. How do you find the inverse of $g(x)$?
3. What is the derivative of $g(x)$?
4. How does shifting $x$ by 3 affect the graph of the cubic function?
5. What is the behavior of $g(x)$ as $x \to \infty$ and $x \to -\infty$?

Tip: For polynomial functions, the domain is almost always all real numbers unless there are division by zero or square roots involved.