Sets, Relations & Functions
Domain of a function
Grade 11
Question:
<p>The domain of the real-valued function \(f(x) = \frac{x-2}{(x-1)\sqrt{x^2-4}}\) is</p>
<p>(a) \((1, 2)\)</p>
<p>(b) \(R - [-2, 2]\)</p>
<p>(c) \((-\infty, -2) \cup (1, +\infty)\)</p>
<p>(d) \(R - [1, 2, -2]\)</p>
Step-by-Step Solution
Key Concept: For the domain, simultaneously satisfy: denominator ≠ 0, expression under square root > 0 (not ≥, since it's in denominator), and x - 2 can be any real number. The √(x² - 4) in denominator requires x² - 4 > 0 strictly.
<p><strong>Step 1:</strong> Identify all restrictions on f(x) = (x-2)/[(x-1)√(x²-4)]</p><p><strong>Step 2:</strong> Restriction 1 – Expression under square root must be positive (not zero since it's in denominator): x² - 4 > 0 ⟹ (x-2)(x+2) > 0 ⟹ x < -2 or x > 2</p><p><strong>Step 3:</strong> Restriction 2 – Denominator cannot be zero: (x-1)√(x²-4) ≠ 0. This means x ≠ 1 AND √(x²-4) ≠ 0. The second part is already covered by Step 2 (strict inequality).</p><p><strong>Step 4:</strong> Check if x = 1 lies in the region from Step 2: Since 1 is in the interval [-2, 2], it's already excluded by x < -2 or x > 2.</p><p><strong>Step 5:</strong> Combine restrictions: Domain = {x ∈ ℝ : x < -2 or x > 2} = (-∞, -2) ∪ (2, ∞)</p><p>∴ Answer: B</p>
Correct Answer: B