Question:
The domain of the function \( f(x) = \frac{\sqrt{9 - x^2}}{\sin^{-1}(3 - x)} \) is ________.
The domain of the function \( f(x) = \frac{\sqrt{9 - x^2}}{\sin^{-1}(3 - x)} \) is ________.
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The domain of the function $f(x)=\sqrt9-x/\sin$ is ____.
Updated On: Apr 15, 2026
- (2, 3)
- [2, 3)
- (2, 3]
- None of these
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Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Square Root Constraint
$9 - x^2 \ge 0 \Rightarrow x^2 \le 9 \Rightarrow -3 \le x \le 3$.
Step 2: Inverse Sine Constraint
$-1 \le 3 - x \le 1 \Rightarrow -4 \le -x \le -2 \Rightarrow 2 \le x \le 4$.
Step 3: Non-zero Denominator
$\sin^{-1}(3-x) \ne 0 \Rightarrow 3 - x \ne 0 \Rightarrow x \ne 3$.
Step 4: Intersection
Intersection of $[-3, 3]$, $[2, 4]$, and $x \ne 3$ is $[2, 3)$.
Final Answer: (B)
$9 - x^2 \ge 0 \Rightarrow x^2 \le 9 \Rightarrow -3 \le x \le 3$.
Step 2: Inverse Sine Constraint
$-1 \le 3 - x \le 1 \Rightarrow -4 \le -x \le -2 \Rightarrow 2 \le x \le 4$.
Step 3: Non-zero Denominator
$\sin^{-1}(3-x) \ne 0 \Rightarrow 3 - x \ne 0 \Rightarrow x \ne 3$.
Step 4: Intersection
Intersection of $[-3, 3]$, $[2, 4]$, and $x \ne 3$ is $[2, 3)$.
Final Answer: (B)
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