Let f(x) = \(x^2\) and g(x) = \(\sin x\) for all \(x \in \mathbb{R}\). Then the set of all \(x\) satisfying (f o g o g o f)(x) = (g o g o f)(x), where (f o g)(x) = f(g(x)), is -
±\(\sqrt{n}\), \(n \in \{0,1,2,...\}\
±\(\sqrt{n}\), \(n \in \{1,2,...\}\
\(\frac{\pi}{2} + 2n\pi\), \(n \in \{..., -2,-1,0,1,2,...\}\
2n\pi, \(n \in \{..., -2,-1,0,1,2,...\}\
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f'(x)=6(x-2)(x-3): not monotone → not one-one. f(0)=1, f(2)=29, f(3)=28 → range=[1,29] → onto.</p><p><strong>Answer: (B) onto but not one-one</strong></p><div class="trap-box"><strong>Trap:</strong> A cubic on a closed interval is not automatically one-one.</div><div class="key-concept"><strong>Key Concept:</strong> Derivative sign + extreme values determines mapping nature on closed interval</div></div>
Correct Answer: A