<p><strong>934.</strong> Find the number of integers not in the domain of \(f(x) = \cos^{-1}\!\left(\dfrac{2 - x}{2x}\right)\).</p>
Step-by-Step Solution
Key Concept: The domain of cos⁻¹(y) requires -1 ≤ y ≤ 1. We need to find all integers x where (2-x)/(2x) falls outside [-1, 1], then count how many integer values of x are excluded.
<p><strong>Step 1:</strong> For f(x) to be defined, we need: -1 ≤ (2-x)/(2x) ≤ 1, with x ≠ 0</p><p><strong>Step 2:</strong> Solve (2-x)/(2x) ≥ -1:</p><p>(2-x)/(2x) + 1 ≥ 0 → (2-x+2x)/(2x) ≥ 0 → (2+x)/(2x) ≥ 0</p><p>This holds when: x < -2 or x > 0</p><p><strong>Step 3:</strong> Solve (2-x)/(2x) ≤ 1:</p><p>(2-x)/(2x) - 1 ≤ 0 → (2-x-2x)/(2x) ≤ 0 → (2-3x)/(2x) ≤ 0</p><p>This holds when: x ≤ 0 or x ≥ 2/3</p><p><strong>Step 4:</strong> Find intersection of both conditions:</p><p>(x < -2 or x > 0) AND (x ≤ 0 or x ≥ 2/3)</p><p>= (x < -2) or (x ≥ 2/3 and x > 0) = x < -2 or x ≥ 2/3</p><p><strong>Step 5:</strong> Integers NOT in domain are those in [-2, 2/3):</p><p>x ∈ {-2, -1, 0}</p><p><strong>Step 6:</strong> Count: -2, -1, 0 are three integers</p><p>∴ Answer: <strong>3</strong></p>
Correct Answer: 3