The given function is:

$g(x) = \frac{x^2}{4 - 3x}$

To determine the domain of this function, we need to identify the values of $x$ for which the function is defined. Specifically, we should find the values of $x$ that make the denominator equal to zero, as division by zero is undefined.

Set the denominator equal to zero and solve for $x$:

$4 - 3x = 0$

$3x = 4$

$x = \frac{4}{3}$

Thus, the function is undefined when $x = \frac{4}{3}$. Therefore, the domain of $g(x)$ includes all real numbers except $x = \frac{4}{3}$.

The domain in interval notation is:

$(-\infty, \frac{4}{3}) \cup (\frac{4}{3}, \infty)$

Would you like further details or explanations on this? Here are some related questions:

1. How do you find the range of a rational function?
2. What happens to the graph of $g(x)$ at $x = \frac{4}{3}$?
3. Can the function have other undefined points besides division by zero?
4. How would the domain change if the numerator also had a variable that could create undefined values?
5. What is the general method for finding domains of more complex rational functions?

Tip: Always check the denominator of rational functions for zeroes to avoid undefined values when determining the domain.