https://file.aiquickdraw.com/mathbot/file/1729648227667jjdxf6gr.jpg

The function given in the image is:

\[
f(x) = \sqrt{\frac{5 - x}{10 - x}}
\]

To compute the domain of this function, we need to ensure two conditions are met:

1. The expression inside the square root must be **non-negative** (since the square root of a negative number is not defined in the set of real numbers).
   
   \[
   \frac{5 - x}{10 - x} \geq 0
   \]

2. The denominator of the fraction must not be zero (since division by zero is undefined).

   \[
   10 - x \neq 0 \quad \Rightarrow \quad x \neq 10
   \]

### Step 1: Solve the inequality
We need to find when:

\[
\frac{5 - x}{10 - x} \geq 0
\]

- First, find the points where the expression is zero or undefined:
  - \( 5 - x = 0 \) implies \( x = 5 \)
  - \( 10 - x = 0 \) implies \( x = 10 \) (which must be excluded because division by zero is undefined).

The critical points are \( x = 5 \) and \( x = 10 \).

Now, check the sign of the expression in the intervals determined by these points:

- For \( x < 5 \), both \( 5 - x > 0 \) and \( 10 - x > 0 \), so \( \frac{5 - x}{10 - x} > 0 \).
- For \( 5 < x < 10 \), \( 5 - x < 0 \) and \( 10 - x > 0 \), so \( \frac{5 - x}{10 - x} < 0 \).
- For \( x > 10 \), both \( 5 - x < 0 \) and \( 10 - x < 0 \), so \( \frac{5 - x}{10 - x} > 0 \).

Thus, the expression is non-negative for \( x \in (-\infty, 5] \cup (10, \infty) \).

### Step 2: Exclude the undefined point
We exclude \( x = 10 \) because the denominator becomes zero at that point. So the domain is:

\[
(-\infty, 5] \cup (10, \infty)
\]

### Conclusion
The correct domain of the function is:

\[
]-\infty, 5] \cup ]10, \infty[
\]

This matches the first option in the image.

---

Would you like more details on any part of this explanation?

Here are 5 related questions you could explore:

1. How do you solve inequalities involving rational expressions in general?
2. What happens if the square root expression includes a negative value?
3. Can a function have more than one point where it is undefined?
4. How can you represent domain restrictions graphically on a number line?
5. How do we handle domains involving complex numbers?

**Tip:** Always check for both non-negative values inside square roots and undefined values due to division by zero when determining domains.

Compute the domain of f(x) = √((5 - x) / (10 - x))

Learn how to find the domain of the function f(x) = √((5 - x) / (10 - x)) with a step-by-step solution. This problem involves solving inequalities for square root functions and rational expressions. Suitable for high school students in Grades 10-12.

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