Question

The domain of the function $\sqrt{\frac{x - 7}{9 - x}}$ is

• A. $(7, 9)$
• B. $[7, 9)$
• C. $[7, 9]$
• D. $(7, 9]$

Hint:

For rational inequalities $\frac{x-a}{x-b} \le 0$, the solution is the bounded interval between roots $[a,b]$. Always check the denominator root separately to ensure it is excluded with an open parenthesis.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For the function \[ f(x) = \sqrt{\frac{x - 7}{9 - x}}, \] the expression inside the square root must be non-negative and the denominator must not be zero.
Step 2: Form the Inequality:
\[ \frac{x - 7}{9 - x} \ge 0 \] Also, \[ 9 - x \neq 0 \;\Rightarrow\; x \neq 9 \]
Step 3: Solve the Inequality:
Rewrite: \[ \frac{x - 7}{9 - x} \ge 0 \;\Rightarrow\; \frac{x - 7}{x - 9} \le 0 \] Critical points: \[ x = 7, x = 9 \] Check sign in intervals:

• $(-\infty,7)$ $\rightarrow$ positive
• $(7,9)$ $\rightarrow$ negative
• $(9,\infty)$ $\rightarrow$ positive

So solution: \[ [7, 9) \]
Step 4: Check Endpoints:
At $x=7$: expression = $0$ (allowed)
At $x=9$: denominator = $0$ (not allowed)
Step 5: Final Answer:
\[ \boxed{[7, 9)} \]