CBSE Class 12 Mathematics Relations and Functions Q #2122
KNOWLEDGE BASED
APPLY
2 Marks Generic-Curriculum VSA
Let $f : R \to R$ be defined as $f(x) = \frac{x^2 + 1}{2}$, then check whether $f$ is one-one or onto.
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Detailed Solution

Step 1: Check for One-One (Injective)

A function is one-one if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. Let $f(x_1) = f(x_2)$. $$ \frac{x_1^2 + 1}{2} = \frac{x_2^2 + 1}{2} $$ $$ x_1^2 = x_2^2 $$ $$ x_1 = \pm x_2 $$ Since $f(1) = f(-1) = 1$, the function is not one-one.

Step 2: Check for Onto (Surjective)

A function is onto if the range equals the codomain ($R$). For $f(x) = \frac{x^2 + 1}{2}$, since $x^2 \ge 0$ for all $x \in R$, we have $x^2 + 1 \ge 1$. Thus, $f(x) \ge \frac{1}{2}$. The range is $[\frac{1}{2}, \infty)$. Since the range is not equal to the codomain $R$, the function is not onto.

Final Answer: The function is neither one-one nor onto.

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the definitions of injective and surjective mappings to a specific algebraic function.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a standard algorithmic process to test the properties of a function defined on real numbers.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE (Textbook). This is a standard textbook problem from the chapter 'Relations and Functions' designed to test foundational understanding of function mapping.

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