Question:

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