To find the domain of the function $g(x) = \frac{2x}{3x - 1}$, we need to determine the values of $x$ for which the function is defined. This function is a rational function, meaning it is defined for all values of $x$ except where the denominator is zero.

Step-by-Step Solution:

1. Identify the denominator of the function:
   The denominator of $g(x)$ is $3x - 1$.

2. Set the denominator equal to zero to find the restricted values:
   $3x - 1 = 0$

3. Solve for $x$: $3x = 1$ $x = \frac{1}{3}$

4. Determine the domain:
   The domain of $g(x)$ is all real numbers except $x = \frac{1}{3}$.

Thus, the domain of $g(x) = \frac{2x}{3x - 1}$ is all real numbers $x$ such that $x \neq \frac{1}{3}$.

Domain in Interval Notation:

The domain in interval notation is: $(-\infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$

Would you like further details or have any questions about this? Here are some related questions:

1. What is the range of the function $g(x) = \frac{2x}{3x - 1}$?
2. How do you find vertical and horizontal asymptotes for rational functions?
3. Can you explain how to determine whether a function is continuous at a given point?
4. What are the steps to find the inverse of a rational function?
5. How do you analyze the behavior of $g(x) = \frac{2x}{3x - 1}$ as $x$ approaches infinity?

Tip: When finding the domain of a rational function, always look for values of $x$ that make the denominator zero, as these are the points where the function is undefined.