The given problem asks to determine the domain of the function $f(x) = \frac{3x}{2x-1}$.

Step-by-Step Solution:

1. Understanding the Domain: The domain of a function consists of all real numbers $x$ for which the function is defined. In the case of a rational function, the denominator must not be zero.

2. Identify the Denominator: The denominator of the function is $2x - 1$. For $f(x)$ to be defined, $2x - 1 \neq 0$.

3. Solve the Equation $2x - 1 = 0$: $2x - 1 = 0 \quad \implies \quad x = \frac{1}{2}$

4. Domain: The function $f(x)$ is defined for all real numbers except $x = \frac{1}{2}$. Therefore, the domain is: $\text{Domain: } x \in \mathbb{R} \setminus \left\{ \frac{1}{2} \right\}$ In interval notation: $(-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

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Related Questions:

1. How do you determine the range of the function $f(x)$?
2. What happens to $f(x)$ as $x \to \infty$?
3. How can the domain be expressed in set notation?
4. Can you calculate $f(x)$ for specific values of $x$, like $x = 1$ or $x = 0$?
5. What is the vertical asymptote of $f(x)$?

Tip: Always check for values that make the denominator zero when dealing with rational functions