Sets, Relations & Functions Question - None

Question

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 $$(f-g)(x)$$ .

Accepted Answer

Key Idea: Write $x\in\mathbb{Q}$ on rational and irrational branches separately, then check cross-branch collisions.

Step 1: $x\in\mathbb{Q}$ : $$(f-g)(x) = 2x+3$$

Step 2: $x\in\mathbb{R}\setminus\mathbb{Q}$ : $(f-g)(x) = 3x-\sqrt{5}$

Step 3: Not one-one — a rational output can equal an irrational-branch output (cross-branch collision exists).

Step 4: Not onto — certain irrationals cannot be achieved from either branch simultaneously.

Answer: Neither one-one nor onto

Trap: Each branch may look injective individually, but cross-branch collisions destroy one-one behaviour.

Key Concept: Rational/irrational piecewise functions — test rational∩irrational image overlap

Question:

Step-by-Step Solution

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