<p>Consider \(f(x) = \dfrac{|x - 4|}{|x| + 1}\). If sum of all distinct possible values of \(\sin^{-1}(\sin[f(x)])\) is \(a\pi + b\) then find the absolute value of \((a + b)\).<br>[<strong>Note:</strong> \([z]\) denotes greatest integer function less than or equal to \(z\).]</p>
Step-by-Step Solution
Key Concept: Find the range of f(x), determine which values of f(x) lead to different greatest integers [f(x)], then apply sin⁻¹(sin[f(x)]) to get distinct values based on where [f(x)] lies relative to [-π/2, π/2].
<p><strong>Step 1: Find the range of f(x) = |x-4|/(|x|+1)</strong></p><p>For x ≥ 4: f(x) = (x-4)/(x+1). As x → ∞, f(x) → 1. At x = 4, f(4) = 0. This function increases from 0 to 1.</p><p>For 0 ≤ x < 4: f(x) = (4-x)/(x+1). At x = 0, f(0) = 4. At x = 4, f(4) = 0. This decreases from 4 to 0.</p><p>For x < 0: f(x) = (4-x)/(-x+1) = (4-x)/(1-x). As x → -∞, f(x) → 1. At x = 0⁻, f(0⁻) = 4/1 = 4.</p><p>The range of f(x) is [0, 4] with maximum value approaching 4.</p><p><strong>Step 2: Determine possible values of [f(x)]</strong></p><p>Since f(x) ∈ [0, 4), the greatest integer function [f(x)] can take values: 0, 1, 2, 3.</p><p>We verify: f(x) = 0 at x = 4; f(x) ∈ [1,2) is achievable; f(x) ∈ [2,3) is achievable; f(x) ∈ [3,4) is achievable.</p><p><strong>Step 3: Apply sin⁻¹(sin[f(x)]) for each case</strong></p><p>• When [f(x)] = 0: sin⁻¹(sin 0) = 0</p><p>• When [f(x)] = 1: sin⁻¹(sin 1) = 1 (since 1 ∈ [-π/2, π/2])</p><p>• When [f(x)] = 2: sin⁻¹(sin 2) = π - 2 (since 2 ∈ [π/2, π], and sin(π-2) = sin 2)</p><p>• When [f(x)] = 3: sin⁻¹(sin 3) = π - 3 (since 3 ∈ [π/2, π], and sin(π-3) = sin 3)</p><p><strong>Step 4: Sum all distinct values</strong></p><p>Sum = 0 + 1 + (π - 2) + (π - 3) = 2π - 4</p><p><strong>Step 5: Express in form aπ + b</strong></p><p>2π - 4 = 2π + (-4)</p><p>Therefore a = 2, b = -4</p><p>|a + b| = |2 + (-4)| = |-2| = 2</p><p><strong>∴ Answer: 2</strong></p>
Correct Answer: 2