CBSE Class 12 Mathematics Relations and Functions Q #2127
KNOWLEDGE BASED
APPLY
3 Marks Generic-Curriculum SA
Show that the function $f : R \to R$ defined by $f(x) = x^2 - 1$ is neither one-one nor onto.
Prev Next

AI Tutor Explanation

Powered by Gemini

Detailed Solution

Step 1: Testing for One-One (Injective)

A function is one-one if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. Let us check for $f(x) = x^2 - 1$. Consider $x_1 = 1$ and $x_2 = -1$. Both are in the domain $R$. $$f(1) = (1)^2 - 1 = 0$$ $$f(-1) = (-1)^2 - 1 = 0$$ Since $f(1) = f(-1)$ but $1 \neq -1$, the function is not one-one.

Step 2: Testing for Onto (Surjective)

A function is onto if for every $y$ in the codomain $R$, there exists an $x$ in the domain $R$ such that $f(x) = y$. Let $y = -2$. $$x^2 - 1 = -2$$ $$x^2 = -1$$ Since $x^2 = -1$ has no real solution, there is no $x \in R$ such that $f(x) = -2$. Thus, the function is not onto.

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

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the formal definitions of injective and surjective mappings to a specific algebraic function.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a standard algorithmic procedure to verify properties of functions using counter-examples.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE (Textbook). This is a foundational problem from the Relations and Functions chapter, essential for building conceptual clarity on mapping types.

More from this Chapter

SA
For $\alpha \in N$, consider a relation $R$ on $N$ given by $R = \{(x, y) : 3x + \alpha y \text{ is a multiple of } 7\}$. If the relation $R$ is an equivalence relation then what can be said about $\alpha$.
VSA
How many equivalence relations on the set {1,2,4} containing (1,2) and (2,1) are there in all? Justify your answer.
LA
Let $N$ be the set of all natural numbers and $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d) \Leftrightarrow ad = bc$ for all $(a, b), (c, d) \in N \times N$. Show that $R$ is an equivalence relation on $N \times N$. Also, find the equivalence class of $(2, 6)$, i.e., $[(2, 6)]$. [All India 2019, 2022]
MCQ_SINGLE
For real x, let \(f(x)=x^{3}+5x+1\). Then:
VSA
Let $R$ be a relation on the set $N$ be defined by $\{(x, y) : x, y \in N, 2x + y = 41\}$. Check whether $R$ is symmetric, reflexive or transitive.
View All Questions