Sets, Relations & Functions
General
Grade 11
Question:
<p>Let \(f(x)=x+3\) for \(x\in\mathbb{Q}\), \(4x\) for \(x\in\mathbb{R}\setminus\mathbb{Q}\); and \(g(x)=\sqrt{5}+x\) for \(x\in\mathbb{R}\setminus\mathbb{Q}\), \(-x\) for \(x\in\mathbb{Q}\). Find the nature of \((f-g)(x)\).</p>
One-one and onto
One-one into
Onto but not one-one
<strong>Neither one-one nor onto</strong>
Step-by-Step Solution
<div class="solution"><p><strong>Key Idea:</strong> Write $(f-g)(x)$ on rational and irrational branches separately, then check cross-branch collisions.</p><p><strong>Step 1:</strong> $x\in\mathbb{Q}$: $(f-g)(x) = 2x+3$</p><p><strong>Step 2:</strong> $x\in\mathbb{R}\setminus\mathbb{Q}$: $(f-g)(x) = 3x-\sqrt{5}$</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 class="key-concept"><strong>Key Concept:</strong> Rational/irrational piecewise functions -- test rational\capirrational image overlap
Correct Answer: 4