<p>If \(f(\ln(1+|x|)) = (1 - \ln(1+|x|))^{\frac{1}{7}}\), then \(f(f(\cos x))\) is equal to:</p>
Step-by-Step Solution
Key Concept: Recognize that f is defined implicitly through a substitution; set t = ln(1+|x|) to find the explicit form f(t), then compose f with itself carefully using the domain constraint that cos x ∈ [-1,1], which limits ln(1+|cos x|) to [0, ln 2].
<p><strong>Step 1:</strong> Find the explicit form of f by substituting t = ln(1+|x|) in the given equation.</p><p>Given: f(ln(1+|x|)) = (1 - ln(1+|x|))^(1/7)</p><p>Let t = ln(1+|x|), then f(t) = (1 - t)^(1/7)</p><p><strong>Step 2:</strong> Determine the domain of t. Since |x| ≥ 0, we have 1+|x| ≥ 1, so ln(1+|x|) ≥ 0. Thus t ∈ [0, ∞).</p><p><strong>Step 3:</strong> Find f(cos x). Since cos x ∈ [-1,1], we have 1+|cos x| ∈ [0, 2], so ln(1+|cos x|) ∈ (-∞, ln 2]. For the principal domain, ln(1+cos x) ∈ [0, ln 2] when cos x ≥ 0.</p><p>Thus f(cos x) = (1 - ln(1+|cos x|))^(1/7)</p><p><strong>Step 4:</strong> Find f(f(cos x)). Let u = f(cos x) = (1 - ln(1+|cos x|))^(1/7).</p><p>Then f(f(cos x)) = f(u) = (1 - u)^(1/7) = (1 - (1 - ln(1+|cos x|))^(1/7))^(1/7)</p><p><strong>Step 5:</strong> Simplify by recognizing the pattern. If we denote α = ln(1+|cos x|), then:</p><p>f(α) = (1 - α)^(1/7)</p><p>f(f(α)) = (1 - (1-α)^(1/7))^(1/7)</p><p>For cos x ∈ [0,1] (when cos x ≥ 0): f(f(cos x)) = (1 - (1 - ln(1+cos x))^(1/7))^(1/7)</p><p>∴ Answer: C</p>
Correct Answer: C