CBSE Class 12 Mathematics Relations and Functions Q #2140
COMPETENCY BASED
APPLY
2 Marks Generic-Curriculum VSA
Let $g(x) = x^2 - 4x - 5$, then check whether $f$ is one-one or onto. [All India 2016]
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Detailed Solution

Step 1: Analyze One-One (Injective) Property

A function $f: A \to B$ is one-one if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. Let us check for $g(x) = x^2 - 4x - 5$. Consider $g(x_1) = g(x_2)$. $$x_1^2 - 4x_1 - 5 = x_2^2 - 4x_2 - 5$$ $$x_1^2 - x_2^2 - 4(x_1 - x_2) = 0$$ $$(x_1 - x_2)(x_1 + x_2) - 4(x_1 - x_2) = 0$$ $$(x_1 - x_2)(x_1 + x_2 - 4) = 0$$ This implies $x_1 = x_2$ or $x_1 + x_2 = 4$. Since there exist distinct values (e.g., $x=1$ and $x=3$ both give $g(1) = -8$ and $g(3) = -8$), the function is not one-one.

Step 2: Analyze Onto (Surjective) Property

A function is onto if its range equals its codomain. Assuming the codomain is $\mathbb{R}$, we find the range of $g(x) = x^2 - 4x - 5$. Completing the square: $$g(x) = (x^2 - 4x + 4) - 4 - 5 = (x - 2)^2 - 9$$ Since $(x - 2)^2 \ge 0$ for all real $x$, the minimum value is $-9$. Thus, the range is $[-9, \infty)$. Since the range is not equal to the codomain $\mathbb{R}$, 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 functions to a specific quadratic algebraic expression.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a standard algorithmic process (algebraic manipulation for one-one and range determination for onto) to verify properties.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the conceptual understanding of Relations and Functions (Chapter 1) beyond rote memorization.

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