<p><strong>703.</strong> Let \(f(x) = \cos^{-1}\!\left(\sqrt{\sin^{-1}\!\left(\sec\!\left(\ln\!\left(\dfrac{2x^2+3x-2}{x^2-3x+2}\right)\right)\right)}\right)\). Find the value of \(1 + \left(\displaystyle\sum \alpha_i^2\right)\), where \(\alpha_i\) represents the integers in the range of \(f(x)\). If there are no integers in the range of \(f(x)\), then enter your answer as zero.</p>
Step-by-Step Solution
Key Concept: First establish the domain constraints from nested inverse functions: sin⁻¹(·) requires input in [-1,1], then √(·) requires non-negative input, then cos⁻¹(·) requires input in [0,1]. These constraints cascade to restrict the logarithmic argument, ultimately determining which integer x-values are valid and what output values f(x) can attain.
<p><strong>Step 1: Simplify the rational expression</strong></p><p>Factor: 2x² + 3x - 2 = (2x - 1)(x + 2) and x² - 3x + 2 = (x - 1)(x - 2)</p><p>So the argument of ln is (2x - 1)(x + 2)/[(x - 1)(x - 2)]</p><p><strong>Step 2: Apply domain constraints</strong></p><p>For sin⁻¹ to be defined: -1 ≤ sec(ln(...)) ≤ 1, which means |cos(ln(...))| ≥ 1</p><p>This requires cos(ln(...)) = ±1, so ln(...) = nπ for integer n</p><p><strong>Step 3: Determine feasible values</strong></p><p>For real x with restricted domain, ln((2x-1)(x+2)/[(x-1)(x-2)]) = 0 (i.e., n = 0)</p><p>This gives (2x - 1)(x + 2) = (x - 1)(x - 2)</p><p>Expanding: 2x² + 3x - 2 = x² - 3x + 2</p><p>So x² + 6x - 4 = 0, giving x = -3 ± √13</p><p>Only x = -3 + √13 ≈ 0.606 is in the domain of the original rational expression</p><p><strong>Step 4: Evaluate f(x)</strong></p><p>When ln(...) = 0: sec(0) = 1, so sin⁻¹(1) = π/2</p><p>Then √(π/2) ≈ 1.253, which exceeds 1 (outside domain of cos⁻¹)</p><p>Checking n = -π: sec(-π) = -1, so sin⁻¹(-1) = -π/2 (negative, can't take square root)</p><p><strong>Step 5: Conclude</strong></p><p>The function has no points in its domain where all nested functions remain real and well-defined simultaneously. Therefore, the range of f(x) contains no integers.</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0