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We are asked to compute the domain of the function

\[
f(x) = \sqrt{\frac{(14 - x)^2}{7 - x}}.
\]

### Domain Constraints:
1. **Square root constraint**: The expression inside the square root must be non-negative, i.e., 
   \[
   \frac{(14 - x)^2}{7 - x} \geq 0.
   \]

2. **Denominator constraint**: The denominator \(7 - x\) cannot be zero, meaning:
   \[
   x \neq 7.
   \]

Let’s analyze these constraints step by step:

### 1. Simplifying the expression:
Since \((14 - x)^2\) is a square term, it is always non-negative. So, we only need to focus on the sign of \(7 - x\).

- For the fraction \(\frac{(14 - x)^2}{7 - x}\) to be non-negative, \(7 - x\) must be positive or negative appropriately.
  
- \(7 - x > 0 \Rightarrow x < 7\). In this case, the entire expression is non-negative because the numerator is non-negative.

### 2. Exclusion of \(x = 7\):
As mentioned, the denominator becomes zero when \(x = 7\), which leads to an undefined expression. So, \(x = 7\) is excluded from the domain.

### 3. Handling \(x = 14\):
At \(x = 14\), \((14 - x)^2 = 0\), making the entire expression equal to 0, which is valid. Therefore, \(x = 14\) is part of the domain.

### Final domain:
- The function is defined for all \(x < 7\) (due to the denominator constraint), and for \(x = 14\).
- The domain is thus:
  \[
  (-\infty, 7) \cup \{14\}.
  \]

### Answer:
The correct domain of the function is \( (-\infty, 7] \cup \{14\} \), which corresponds to the first option in the list:
\[
] -\infty, 7] \cup \{14\}.
\]

---

Would you like further details on any of these steps? Here are five related questions you could explore:

1. How do we handle other types of square root functions with different polynomials in the numerator?
2. What if the numerator of the expression had a different sign pattern?
3. How does the domain change if the exponent on \(14 - x\) is odd rather than even?
4. Can you give an example of a function where both the numerator and denominator contribute equally to domain restrictions?
5. How does the addition of complex numbers impact the domain of square root functions?

**Tip:** Always double-check both the numerator and denominator when computing domains for rational functions!

Compute the domain of the function f(x) = sqrt[((14 - x)^2) / (7 - x)].

Learn how to compute the domain of the function f(x) = sqrt[((14 - x)^2) / (7 - x)] by analyzing the square root and denominator constraints. This problem covers algebraic concepts such as inequalities and rational functions, suitable for high school students.

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