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.