<p>The domain of the function \( f(x) = \sin^{-1}\left[\log_3\left(\dfrac{x}{3}\right)\right] \) is:</p>
Step-by-Step Solution
Key Concept: For sin⁻¹ to be defined, its argument must lie in [-1,1]. Simultaneously, the logarithm requires its argument to be positive. You must solve both constraints: -1 ≤ log₃(x/3) ≤ 1 AND x/3 > 0.
<p><strong>Step 1:</strong> For sin⁻¹[log₃(x/3)] to be defined, we need:</p><ul><li>x/3 > 0 (for logarithm to exist) ⟹ x > 0</li><li>-1 ≤ log₃(x/3) ≤ 1 (for sin⁻¹ to be defined)</li></ul><p><strong>Step 2:</strong> Solve -1 ≤ log₃(x/3) ≤ 1</p><p>Left inequality: log₃(x/3) ≥ -1</p><p>3⁻¹ ≤ x/3</p><p>1/3 ≤ x/3</p><p>x ≥ 1</p><p><strong>Step 3:</strong> Right inequality: log₃(x/3) ≤ 1</p><p>x/3 ≤ 3¹</p><p>x/3 ≤ 3</p><p>x ≤ 9</p><p><strong>Step 4:</strong> Combine all constraints: x > 0, x ≥ 1, and x ≤ 9</p><p>∴ Domain: <strong>[1, 9]</strong></p>
Correct Answer: A