Select the correct answer.
A group of 6 students is surveying their classmates to get feedback about the different clubs they’re involved in. If each student surveys more than 2 of their classmates, the total number of completed surveys, s(x)
, is given by the equation s(x)\:=\:6x
.
Which could be the domain of this function?  A. \left \{ x\:\epsilon \:\mathbb{R}\:|\:x\:>\:2 \right \}
 B. \left \{ x\:\epsilon \:\mathbb{Q}\:|\:x\:\geq \:2 \right \}
 C. \left \{ x\:\epsilon \:\mathbb{N}\:|\:x\:>\:2 \right \}
 D. \left \{ x\:\epsilon \:\mathbb{Z}\:|\:x\:\geq \:2 \right \}


To determine the correct domain for the function \( s(x) = 6x \), we need to consider the context of the problem:

- **\( x \)** represents the number of classmates each student surveys.
- The number of classmates surveyed must be more than 2.

Now, let's analyze the options:

1. **Option A: \( \left\{ x \in \mathbb{R} \mid x > 2 \right\} \)**
   - \( \mathbb{R} \) represents the set of all real numbers.
   - This would imply that \( x \) can be any real number greater than 2, including non-integer values (like 2.5, 3.7, etc.).
   - However, the number of classmates must be a whole number (integer), so this option is **not correct**.

2. **Option B: \( \left\{ x \in \mathbb{Q} \mid x \geq 2 \right\} \)**
   - \( \mathbb{Q} \) represents the set of all rational numbers.
   - Similar to option A, this would allow non-integer values for \( x \), which doesn't make sense in this context. Therefore, this option is **not correct**.

3. **Option C: \( \left\{ x \in \mathbb{N} \mid x > 2 \right\} \)**
   - \( \mathbb{N} \) represents the set of natural numbers (positive integers).
   - This means \( x \) must be a natural number greater than 2 (like 3, 4, 5, etc.).
   - Since the number of classmates surveyed must be more than 2 and a whole number, this option is **correct**.

4. **Option D: \( \left\{ x \in \mathbb{Z} \mid x \geq 2 \right\} \)**
   - \( \mathbb{Z} \) represents the set of all integers.
   - This includes negative numbers and zero, which are not suitable for this context.
   - Although \( x \geq 2 \) limits \( x \) to 2 and above, the function specifically requires \( x \) to be more than 2, so this option is **not the best fit**.

### Final Answer:
- **Option C** \( \left\{ x \in \mathbb{N} \mid x > 2 \right\} \) is the correct choice for the domain.

Let me know if you need further clarification or have any other questions!

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Here are 5 related questions:
1. What would the domain be if each student could survey exactly 2 classmates?
2. How would the domain change if \( x \) had to be an integer greater than 3?
3. Why can't \( x \) be a real number in this context?
4. How does the domain relate to the possible number of completed surveys?
5. What happens if the function were defined as \( s(x) = 6x + 1 \)?

**Tip:** Always consider the context when determining the domain of a function, especially when dealing with real-world scenarios that involve counts or measurements.

Understand how to determine the domain of a function s(x) = 6x through a detailed explanation. This problem involves identifying suitable values for x based on real-world survey constraints. Suitable for high school students studying functions and domain concepts.

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