CBSE Class 12 Mathematics Relations and Functions Q #2139
COMPETENCY BASED
APPLY
2 Marks Generic-Curriculum VSA
Check whether the mapping $f : N \to N$ given by $f(n) = 1 + n^2, n \in N$ when $N$ is the set of natural numbers, is one-one or onto. [Competency]
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Detailed Solution

Step 1: Check for One-One (Injective) mapping

A function $f$ is one-one if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. Let $f(n_1) = f(n_2)$. $$1 + n_1^2 = 1 + n_2^2$$ $$n_1^2 = n_2^2$$ Since $n_1, n_2 \in N$ (natural numbers are positive), we have $n_1 = n_2$. Thus, the function is one-one.

Step 2: Check for Onto (Surjective) mapping

A function is onto if for every $y \in N$ (codomain), there exists an $x \in N$ (domain) such that $f(x) = y$. Let $f(n) = y$, so $1 + n^2 = y$, which gives $n^2 = y - 1$ or $n = \sqrt{y - 1}$. If we take $y = 2$, then $n = \sqrt{2 - 1} = 1 \in N$. However, if we take $y = 3$, then $n = \sqrt{3 - 1} = \sqrt{2}$, which is not a natural number. Since there exists $y \in N$ that has no pre-image in $N$, the function is not onto.

Final Answer: The function is one-one but not onto.

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the formal definitions of injective and surjective functions to a specific algebraic rule.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a standard algorithmic procedure to test the properties of a function mapping.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the conceptual understanding of Relations and Functions (Chapter 1) beyond rote memorization.

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