The given function is \( y = \frac{1}{3x} + 2 \).

To find the domain of a rational function, we need to determine the values of \( x \) for which the function is defined. A rational function is undefined where the denominator is zero because division by zero is undefined.

### Steps to determine the domain:

1. **Identify the denominator:** 
   The denominator in your function is \( 3x \).
   
2. **Set the denominator equal to zero and solve:**
   \[
   3x = 0
   \]
   \[
   x = 0
   \]
   The function is undefined when \( x = 0 \).

3. **Determine the domain:**
   The domain of the function is all real numbers except where the denominator is zero. Therefore, the domain is:
   \[
   \text{Domain: } x \in \mathbb{R} \text{ and } x \neq 0
   \]
   In interval notation, this is written as:
   \[
   (-\infty, 0) \cup (0, \infty)
   \]

Would you like further details or have any questions about this? Here are five related questions you might find helpful:

1. How do you find the range of a rational function?
2. What is the domain of the function \( y = \frac{2x + 1}{x - 5} \)?
3. How do vertical asymptotes relate to the domain of a function?
4. What is the difference between a hole and an asymptote in a rational function?
5. How do you determine if a function is continuous?

**Tip:** Always check the denominator of a rational function to ensure you understand where the function is undefined. This will help you accurately determine the domain.

Learn how to find the domain of the rational function y = 1/(3x) + 2. Understand the steps to determine where the function is defined, including identifying the denominator, setting it equal to zero, and finding the domain in interval notation. Suitable for high school students studying functions and algebra.

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