<p>The domain of the function \(f(x) = \dfrac{\sin^{-1}(x-3)}{\sqrt{9-x^2}}\) is</p>
Step-by-Step Solution
Key Concept: The domain requires satisfying TWO constraints simultaneously: sin⁻¹(x-3) needs -1 ≤ x-3 ≤ 1, AND the denominator √(9-x²) needs 9-x² > 0 (strict inequality since denominator cannot be zero).
<p><strong>Step 1: Constraint from sin⁻¹(x-3)</strong></p><p>For sin⁻¹(x-3) to be defined:</p><p>-1 ≤ x-3 ≤ 1</p><p>2 ≤ x ≤ 4</p><p><strong>Step 2: Constraint from denominator √(9-x²)</strong></p><p>For √(9-x²) to be defined AND non-zero:</p><p>9-x² > 0 (strict inequality)</p><p>x² < 9</p><p>-3 < x < 3</p><p><strong>Step 3: Find intersection of both constraints</strong></p><p>[2, 4] ∩ (-3, 3) = [2, 3)</p><p>The left endpoint x=2 is included (satisfies both), but x=3 is excluded (makes denominator zero).</p><p>∴ Domain: <strong>[2, 3)</strong></p>
Correct Answer: B