<p>The domain of definition of the function \(f(x) = \sqrt{\log_{10}\left(\dfrac{5x - x^2}{4}\right)}\) is</p>
Step-by-Step Solution
Key Concept: For f(x) to be defined, we need the argument of the square root to be non-negative AND the argument of the logarithm to be positive. This gives us two conditions: (5x - x²)/4 > 0 and log₁₀((5x - x²)/4) ≥ 0.
<p><strong>Step 1:</strong> For the square root to be defined, we need: log₁₀((5x - x²)/4) ≥ 0</p><p><strong>Step 2:</strong> log₁₀(y) ≥ 0 means y ≥ 10⁰ = 1, so: (5x - x²)/4 ≥ 1</p><p><strong>Step 3:</strong> Simplify: 5x - x² ≥ 4 → -x² + 5x - 4 ≥ 0 → x² - 5x + 4 ≤ 0</p><p><strong>Step 4:</strong> Factor: (x - 1)(x - 4) ≤ 0</p><p><strong>Step 5:</strong> This inequality holds when x lies between the roots: 1 ≤ x ≤ 4</p><p><strong>Step 6:</strong> Verify the expression (5x - x²)/4 is positive in this interval: at x = 2: (10 - 4)/4 = 1.5 > 0 ✓</p><p>∴ Domain: [1, 4]</p>
Correct Answer: A