The function provided is $y = \frac{1}{5x}$.

Analyzing the Function:

To find the domain of the function, we need to determine the values of $x$ for which the function is defined.

• The function involves a division by $5x$.
• Division by zero is undefined, so the function is undefined when $5x = 0$.

This implies: $5x = 0 \Rightarrow x = 0$

Therefore, the function is undefined at $x = 0$. For all other values of $x$, the function is defined.

Domain:

The domain of the function $y = \frac{1}{5x}$ includes all real numbers except $x = 0$.

In interval notation, the domain is: $(-\infty, 0) \cup (0, \infty)$

Correct Answer:

The correct option is:

Option C: $(-\infty, 0) \cup (0, \infty)$

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

1. How do you find the domain of a rational function?
2. What happens to the function $y = \frac{1}{5x}$ as $x$ approaches zero?
3. How do you express the domain of a function using interval notation?
4. Can a function have a domain that is all real numbers? Provide an example.
5. What is the difference between the domain and range of a function?
6. How do vertical asymptotes relate to the domain of a function?
7. How would the domain change if the function were $y = \frac{1}{5(x-2)}$?
8. What is the effect on the domain if a function is squared, such as $y = \frac{1}{(5x)^2}$?

Tip: Always watch out for division by zero or taking the square root of a negative number when determining the domain of a function.