Question

The domain of the function \( f(x) = \frac{\sin^{-1} \left( x-3 \right)}{\sqrt{9 - x^2}} \) is:

• A. \( [1,2] \)
• B. \( [2,3] \)
• C. \( [2,3) \)
• D. \( [1,2) \)
• E. \( (1,2) \)

Hint:

Ensure both the argument of the inverse sine function and the square root conditions are satisfied to determine the correct domain.

Answer 1

The Correct Option is C

To find the domain of \( f(x) \), we need to ensure that the argument of the inverse sine function, \( x-3 \), lies within the interval \([-1,1]\) and that the denominator, \( \sqrt{9 - x^2} \), remains non-zero and real. 
1. Argument of \( \sin^{-1} \) within \([-1,1]\): - The condition \( -1 \leq x-3 \leq 1 \) simplifies to: \[ 2 \leq x \leq 4 \] 
2. Denominator non-zero and real: - The square root \( \sqrt{9 - x^2} \) is defined and non-zero when: \[ 0 < x^2 < 9 \] 
- This translates to: \[ -3 < x < 3 \] 
3. Intersection of Conditions: - The intersection of \( 2 \leq x \leq 4 \) and \( -3 < x < 3 \) is: \[ 2 \leq x < 3 \] 
4. Conclusion: - Therefore, the domain of \( f(x) \) is \( [2, 3) \), where \( x \) starts at 2 and approaches but does not include 3.