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The problem gives the set \( K = \{(q, p) \mid q \in \mathbb{R}, p \in \mathbb{R}, p = \sqrt{7 - q} \} \) and asks for the domain and range of \( K \).

### 1. **Domain**:
The domain refers to the possible values of \( q \). Since \( p = \sqrt{7 - q} \), we must have \( 7 - q \geq 0 \) for the square root to be defined (because the square root function is only defined for non-negative numbers in the real number system).

Thus, solving:
\[
7 - q \geq 0 \implies q \leq 7
\]
Therefore, the domain of \( q \) is:
\[
q \in (-\infty, 7]
\]

### 2. **Range**:
The range refers to the possible values of \( p \). Since \( p = \sqrt{7 - q} \), the values of \( p \) depend on the values of \( q \). As \( q \) approaches its maximum value of 7, \( p = \sqrt{7 - q} = \sqrt{0} = 0 \). The minimum value of \( q \) can be \( -\infty \), but the maximum value that \( 7 - q \) can reach is 7, so the largest value of \( p \) is \( \sqrt{7} \).

Thus, the range of \( p \) is:
\[
p \in [0, \sqrt{7}]
\]

### Final Answer:
- **Domain**: \( (-\infty, 7] \)
- **Range**: \( [0, \sqrt{7}] \)

Would you like further clarification or details? Here are some related questions to explore:

1. How would the domain change if \( p = \sqrt{q - 7} \)?
2. What would be the domain and range if the function were \( p = \sqrt{q + 7} \)?
3. How do we determine the domain and range for general square root functions?
4. Can we express this function as \( p = f(q) \), and what kind of transformation does it represent?
5. What would the graph of \( p = \sqrt{7 - q} \) look like in the coordinate plane?

**Tip**: Always start by checking the domain restrictions imposed by square roots or logarithms to determine where the function is defined.

Given that K = {(q,p) | q ∈ R, p ∈ R, p = √(7 − q)}. Determine the Domain and Range of K.

Learn how to determine the domain and range of the function p = √(7 − q). This example covers the application of square root functions, domain restrictions, and finding ranges. Suitable for students in Grades 9-12.

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