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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.

What is the domain of g(x) = x^2 / (4 - 3x)?

Learn how to find the domain of the rational function g(x) = x^2 / (4 - 3x). This problem covers key algebraic concepts, including identifying where the function is undefined and applying the rule of division by zero. Suitable for students in Grades 9-11.

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