Question:

<div class="solution"><p><strong>Step 1:</strong> $g(x)=-x^2+2x+3$, vertex at $x=1$, max value 4. So $g(x)\in(0,4]$ (domain of log).</p><p><strong>Step 2:</strong> $\log_2 g(x)\in(-\infty,2]$.</p><p><strong>Step 3:</strong> $\sin^{-1}$ clips input to $[-1,1]$, so output range is $[\sin^{-1}(-1),\sin^{-1}(1)]=[-\pi/2,\pi/2]$.</p><p><strong>Answer: (A) $[-\pi/2,\,\pi/2]$</strong></p><div class="trap-box"><strong>Trap:</strong> Confusing the range of the inner function with the final range. $\sin^{-1}$ clips the input but covers its full output range.<div class="key-concept"><strong>Key Concept:</strong> Range chain: inner range \to intersect with outer domain \to map to output