To determine the domain of the function $g(x) = \frac{\sqrt{9 - x}}{x - 10}$, we need to consider two conditions:

1. Square root restriction: The expression inside the square root, $9 - x$, must be non-negative, as square roots of negative numbers are not real. Therefore, we need: $9 - x \geq 0$ Solving this inequality: $x \leq 9$

2. Denominator restriction: The denominator $x - 10$ must not be zero, since division by zero is undefined. Thus, we require: $x \neq 10$

Combining the conditions:

• From the square root condition: $x \leq 9$
• From the denominator condition: $x \neq 10$

The domain of the function is all values of $x$ that satisfy $x \leq 9$ but exclude $x = 10$. Therefore, the domain is: $D(g) = (-\infty, 9] \setminus \{10\}$

Would you like more details or further clarification on these steps?

Here are 5 related questions to consider:

1. How do you solve inequalities involving square roots?
2. What happens when a function has multiple restrictions for its domain?
3. How can discontinuities like holes or vertical asymptotes affect a function's graph?
4. What are the general steps for finding the domain of a rational function?
5. What are the differences between open and closed intervals in domains?

Tip: Always check both the numerator and denominator when finding the domain of rational functions!