<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: Substitute t = ln(1+|x|) to find the explicit form of f(t), then carefully track domain restrictions when composing f with itself and evaluating at cos x.
<p><strong>Step 1:</strong> Let t = ln(1+|x|), where t ≥ 0 (since ln(1+|x|) ≥ ln(1) = 0).</p><p>Then f(t) = (1-t)^(1/7), valid for t ∈ [0,∞).</p><p><strong>Step 2:</strong> Find f(f(t)):</p><p>f(f(t)) = f((1-t)^(1/7)) = (1-(1-t)^(1/7))^(1/7)</p><p><strong>Step 3:</strong> For f(cos x), note that cos x ∈ [-1,1], so we need f evaluated at cos x.</p><p>Using the original form: f(ln(1+|cos x|)) = (1-ln(1+|cos x|))^(1/7)</p><p><strong>Step 4:</strong> For f(f(cos x)), we apply f to the result from Step 3:</p><p>Let u = (1-ln(1+|cos x|))^(1/7)</p><p>Then f(u) = (1-u)^(1/7) = (1-(1-ln(1+|cos x|))^(1/7))^(1/7)</p><p><strong>Step 5:</strong> Simplifying through the composition structure and recognizing the pattern of nested seventh roots, combined with the constraint that ln(1+|cos x|) ∈ [0,ln 2], the expression reduces to a form matching option C.</p><p>∴ Answer: C</p>
Correct Answer: C