CBSE Class 12 Mathematics Relations and Functions Q #2129
COMPETENCY BASED
APPLY
3 Marks Generic-Curriculum SA
Prove that the greatest integer function $f : R \to R$ given by $f(x) = [x]$, is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to $x$. [Competency]
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Detailed Solution

Step 1: Testing for One-One (Injective)

A function is one-one if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. Consider two distinct real numbers, for example, $x_1 = 1.2$ and $x_2 = 1.5$. Here, $f(1.2) = [1.2] = 1$ and $f(1.5) = [1.5] = 1$. Since $f(1.2) = f(1.5)$ but $1.2 \neq 1.5$, the function is not one-one.

Step 2: Testing for Onto (Surjective)

A function is onto if the range equals the codomain. The codomain is given as $R$. However, the range of the greatest integer function $f(x) = [x]$ is the set of all integers, $Z$. Since $Z \subset R$ and $Z \neq R$, there exist elements in the codomain (e.g., $0.5$) that have no pre-image in the domain $R$ such that $[x] = 0.5$. Thus, 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 formal definitions of injectivity and surjectivity to a specific mathematical function.
Knowledge Dimension: CONCEPTUAL
Justification: The student must understand the underlying concepts of domain, codomain, and range mapping to evaluate the properties of the function.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the student's ability to reason through abstract definitions rather than rote memorization of function properties.

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