Question:

<div class="solution"><p><strong>Step 1:</strong> Domain = intersection of domains of all three terms.</p><p>• \(\sin^{-1}x\): \([-1,1]\)<br>• \(\tan^{-1}x\): \(\mathbb{R}\)<br>• \(\sec^{-1}x\): \((-\infty,-1]\cup[1,\infty)\)</p><p>Intersection: \(\{-1,1\}\) only.</p><p><strong>Step 2:</strong> <span class="math-block">\[f(1)=\frac{\pi}{2}+\frac{\pi}{4}+0=\frac{3\pi}{4}\]<span class="math-block">\[f(-1)=-\frac{\pi}{2}-\frac{\pi}{4}+\pi=\frac{\pi}{4}\]</p><p><strong>Answer: (C) \(\{\pi/4,\,3\pi/4\}\)</strong> (two discrete values)</p><div class="trap-box"><strong>Trap:</strong> Assuming the range is a continuous interval. The domain is only two points.<div class="key-concept"><strong>Key Concept:</strong> Identify domain first -- a discrete domain gives a discrete range