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!