MCQMathematics

NCERT · Class 11 · Mathematics · Relations and FunctionsThe function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x$ is:

Q: The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x$ is:

To check injectivity: if $f(x1) = f(x2)$, then $2x1 = 2x2 \implies x1 = x2$, so $f$ is one-one (injective). To check surjectivity: for any $y \in \mathbb{R}$, there exists $x = y/2 \in \mathbb{R}$ such that $f(x) = 2(y/2) = y$, so $f$ is onto (surjective). Thus, $f$ is both one-one and onto (bijective). Option A is correct.

Step-by-Step Solution

To check injectivity: if $f(x_1) = f(x_2)$, then $2x_1 = 2x_2 \implies x_1 = x_2$, so $f$ is one-one (injective). To check surjectivity: for any $y \in \mathbb{R}$, there exists $x = y/2 \in \mathbb{R}$ such that $f(x) = 2(y/2) = y$, so $f$ is onto (surjective). Thus, $f$ is both one-one and onto (bijective). Option A is correct.

Detailed Options Breakdown

Option : One-one and Onto (Correct Answer)

Correct choice. Refer to the step-by-step verified solution guidelines above for details.

Option 1: One-one but not Onto

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Relations and Functions.

Option 2: Onto but not One-one

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Relations and Functions.

Option 3: Neither One-one nor Onto

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Relations and Functions.

💡 Study Guide: This question tests core syllabus concepts from Relations and Functions. For formulas, key summaries, and mock exam reference guides, read the full Relations and Functions Revision Notes.

✓

Verified by Academic Committee

← All Chapter Questions Mathematics Chapters