The range of the function f(x) = \text{sgn}\left( \frac{\sin^{2} x + 2\sin x + 4}{\sin^{2} x + 2\sin x + 3} \right) is (where \text{sgn}(.) denotes signum function)-
Step-by-Step Solution
Key Concept: Write $(f-g)(x)$ on rational and irrational branches separately, then check cross-branch collisions.
<div class="solution"><p><strong>Key Idea:</strong> Write <span class="math-inline">$(f-g)(x)$</span> on rational and irrational branches separately, then check cross-branch collisions.</p><p><strong>Step 1:</strong> <span class="math-inline">$x\in\mathbb{Q}$</span>: <span class="math-inline">$(f-g)(x) = 2x+3$</span></p><p><strong>Step 2:</strong> <span class="math-inline">$x\in\mathbb{R}\setminus\mathbb{Q}$</span>: <span class="math-inline">$(f-g)(x) = 3x-\sqrt{5}$</span></p><p><strong>Step 3:</strong> Not one-one — a rational output can equal an irrational-branch output (cross-branch collision exists).</p><p><strong>Step 4:</strong> Not onto — certain irrationals cannot be achieved from either branch simultaneously.</p><p><strong>Answer: Neither one-one nor onto</strong></p><div class="trap-box"><strong>Trap:</strong> Each branch may look injective individually, but cross-branch collisions destroy one-one behaviour.</div><div class="key-concept"><strong>Key Concept:</strong> Rational/irrational piecewise functions — test rational∩irrational image overlap</div></div>
Correct Answer: C