Question:

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