Question:

<div class="solution"><p><strong>Step 1:</strong> \(\sqrt{x}\) requires \(x\ge 0\).</p><p><strong>Step 2:</strong> \(\{\sqrt{x}\}\in[0,1)\) for all \(x\ge 0\) -- always in domain of \(\cos^{-1}\).</p><p><strong>Step 3:</strong> \(\cos^{-1}\{\sqrt{x}\}>0\) always (since \(\{\sqrt{x}\}<1\) so argument \(\ne 1\), and \(\cos^{-1}(0)=\pi/2>0\)).</p><p><strong>Answer: (B) \(x\in[0,\infty)\)</strong></p><div class="trap-box"><strong>Trap:</strong> Thinking integers are excluded because \(\{\sqrt{x}\}=0\); but \(\cos^{-1}(0)=\pi/2>0\), so the log is still defined.<div class="key-concept"><strong>Key Concept:</strong> Range of \(\{\cdot\}\) is \([0,1)\); \(\cos^{-1}\) of anything in \([0,1)\) is positive