The number of integers lying in the domain of the function f(x) = <span class="math-inline">\(\sqrt{\frac{5-2x}{x}}\)</span> is -
Step-by-Step Solution
Key Concept: For f(x) = √[(5-2x)/x] to be defined, the expression under the square root must be non-negative AND the denominator cannot be zero. This requires (5-2x)/x ≥ 0, which involves analyzing the sign of both numerator and denominator across critical points.
<p><strong>Step 1: Identify restrictions</strong></p><p>For f(x) = √[(5-2x)/x] to be defined:</p><p>(i) The radicand must be ≥ 0: (5-2x)/x ≥ 0</p><p>(ii) Denominator ≠ 0: x ≠ 0</p><p><strong>Step 2: Find critical points</strong></p><p>Numerator: 5-2x = 0 ⟹ x = 5/2 = 2.5</p><p>Denominator: x = 0</p><p><strong>Step 3: Sign analysis of (5-2x)/x</strong></p><p>Create a sign table with critical points 0 and 2.5:</p><p>• When x < 0: numerator (5-2x) > 0, denominator x < 0 ⟹ fraction < 0 ✗</p><p>• When 0 < x < 2.5: numerator (5-2x) > 0, denominator x > 0 ⟹ fraction > 0 ✓</p><p>• When x = 2.5: fraction = 0 ✓</p><p>• When x > 2.5: numerator (5-2x) < 0, denominator x > 0 ⟹ fraction < 0 ✗</p><p><strong>Step 4: Determine domain</strong></p><p>Domain: 0 < x ≤ 2.5 or x ∈ (0, 2.5]</p><p><strong>Step 5: Count integers in domain</strong></p><p>Integers in (0, 2.5]: only x = 1 and x = 2</p><p>Check: x = 1: (5-2)/1 = 3 ≥ 0 ✓</p><p>Check: x = 2: (5-4)/2 = 1/2 ≥ 0 ✓</p><p>x = 3 is outside domain since 3 > 2.5</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C