<p>Let <em>f</em>(<em>x</em>) = |<em>x</em> − 2| and <em>g</em>(<em>x</em>) = \(\underbrace{f(f(f(f\ldots(f(x)))\ldots))}_{n\text{ time}}\). If the equation for, \(k \in (0,2)\) then</p>
<p>(a) \(g(x) = k\) has 8 distinct solutions then \(n = 4\)</p>
<p>(b) \(g(x) = k\) has 11 distinct solutions then \(n = 5\)</p>
<p>(c) \(g(x) = k\) has 6 distinct solutions then \(n = 3\)</p>
<p>(d) \(g(x) = k\) has 7 distinct solutions then \(n = 3\)</p>
Step-by-Step Solution
Key Concept: For f(x) = |x - 2|, successive iterations shift the distance from 2 by a factor related to the piecewise structure of absolute value. Understanding that g_n(x) creates nested V-shaped functions is crucial: g_n(x) = |...||x - 2| - 2| - 2...| with n layers, which produces 2^n linear pieces.
<p><strong>Step 1:</strong> Analyze the first few iterations of f(x) = |x - 2|</p><p>• g₁(x) = |x - 2|</p><p>• g₂(x) = ||x - 2| - 2| with breakpoints at x = 0, 2, 4</p><p>• g₃(x) = |||x - 2| - 2| - 2| creates 2³ = 8 linear pieces</p><p><strong>Step 2:</strong> For k ∈ (0, 2), determine the structure of g_n(x)</p><p>Each iteration of f creates a piecewise linear function. The equation g_n(x) = k for k ∈ (0, 2) will have solutions at each "valley" region. As n increases, the function oscillates more rapidly.</p><p><strong>Step 3:</strong> Key observations about solutions for k ∈ (0, 2)</p><p>• At each iteration level, the minimum value occurs at specific points and equals 0 (when nested absolute values equal 2)</p><p>• For g₁(x) = k: solutions exist (2 solutions for k ∈ (0, 2))</p><p>• For g_n(x) = k with n ≥ 1: the number of solutions = 2ⁿ when k ∈ (0, 2)</p><p>• The horizontal line y = k intersects the zigzag pattern of g_n(x) exactly 2ⁿ times</p><p><strong>Step 4:</strong> Verify properties for k ∈ (0, 2)</p><p>A) Number of solutions to g_n(x) = k is 2ⁿ ✓</p><p>B) All solutions lie in a finite bounded interval (related to iterations from starting domain)</p><p>C) Solutions are symmetric about certain critical points determined by the structure of iteration</p><p>∴ Answer: A, B, C</p>
Correct Answer: A,B,C