Sets, Relations & Functions
Domain of a function
Grade 11
Question:
<p>The domain of the definition of the function \(f(x) = \dfrac{1}{4-x^2} + \log_{10}(x^3 - x)\) is:</p>
<p>\((-1, 0) \cup (1, 2) \cup (3, \infty)\)</p>
<p>\((-2, -1) \cup (-1, 0) \cup (2, \infty)\)</p>
<p>\((-1, 0) \cup (1, 2) \cup (2, \infty)\)</p>
<p>\((1, 2) \cup (2, \infty)\)</p>
Step-by-Step Solution
Key Concept: For f(x) to be defined, BOTH conditions must hold simultaneously: (1) denominator 4-x² ≠ 0, AND (2) argument x³-x must be strictly positive for the logarithm. The domain is the intersection of these constraints.
<p><strong>Step 1: Condition from denominator</strong></p><p>For 1/(4-x²) to be defined: 4-x² ≠ 0 ⟹ x ≠ ±2</p><p><strong>Step 2: Condition from logarithm</strong></p><p>For log₁₀(x³-x) to be defined: x³-x > 0</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: Find intersection</strong></p><p>Domain = [(-1, 0) ∪ (1, ∞)] ∩ (ℝ \ {-2, 2})</p><p>Since -2 and 2 don't lie in (-1, 0) ∪ (1, ∞), no additional exclusion occurs from the first condition.</p><p>∴ Domain: <strong>(-1, 0) ∪ (1, ∞)</strong> [Answer: C]</p>
Correct Answer: C