Question:

<div class="solution"><p><strong>Key Idea:</strong> The natural domain is [2,4] but f rises then falls. Restrict to one monotone side.</p><p><strong>Step 1:</strong> <span class="math-inline">\(f'(x) = \frac{1}{2\sqrt{x-2}} - \frac{1}{2\sqrt{4-x}}\)</span> — positive on [2,3), negative on (3,4]. So f is strictly decreasing on [3,4].</p><p><strong>Step 2:</strong> On [3,4], range is <span class="math-inline">\([f(4),f(3)] = [\sqrt{2}, 2]\)</span></p><p><strong>Answer: <span class="math-inline">\(X=[3,4],\ Y=[\sqrt{2},2]\)</span></strong></p><div class="trap-box"><strong>Trap:</strong> Full domain [2,4] gives correct range but f(2)=f(4), so not injective.</div><div class="key-concept"><strong>Key Concept:</strong> Restrict domain to monotone branch, then match codomain to that branch's range</div></div>