https://file.aiquickdraw.com/mathbot/file/1726587196374vzr710s0.jpg

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!

Déterminer le domaine de la fonction g(x) = (√(9 - x)) / (x - 10)

Learn how to find the domain of the function g(x) = √(9 - x) / (x - 10), including solving restrictions on square roots and rational functions. This problem is ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation