<p>The domain of the function \( f(x) = \dfrac{1}{4-x^2} + \log_{10}(x^3 - x) \) is</p>
<p>\((-1, 0) \cup (1, 2) \cup (2, \infty)\)</p>
<p>\((-1, 0) \cup (1, \infty)\)</p>
<p>\((1, 2) \cup (2, \infty)\)</p>
<p>\((-\infty, -1) \cup (1, \infty)\)</p>
Step-by-Step Solution
Key Concept: The domain requires BOTH denominators non-zero AND the logarithm argument positive. You must find the intersection of conditions: 4-x²≠0, x³-x>0 (not ≥0, strictly greater).
<p><strong>Step 1:</strong> From the fraction 1/(4-x²): We need 4-x²≠0, so x²≠4, giving x≠±2</p><p><strong>Step 2:</strong> From log₁₀(x³-x): We need x³-x>0 (strictly positive for logarithm)</p><p>Factor: x(x²-1)>0 ⟹ x(x-1)(x+1)>0</p><p>Sign analysis: x∈(-1,0)∪(1,∞)</p><p><strong>Step 3:</strong> Find intersection of conditions:</p><p>• x≠±2 (from fraction)</p><p>• x∈(-1,0)∪(1,∞) (from logarithm)</p><p>The interval (-1,0) contains no excluded points ±2</p><p>The interval (1,∞) contains x=2, so we must exclude it: (1,2)∪(2,∞)</p><p><strong>Step 4:</strong> Combine both valid regions:</p><p>∴ Domain = (-1,0)∪(1,2)∪(2,∞)</p>
Correct Answer: A