Class CBSE Class 12 Mathematics Relations and Functions Q #1770
COMPETENCY BASED
APPLY
3 Marks 2026 AISSCE(Board Exam) SA
Let $A=\mathbb{R}-\{3\}$ and $B=\mathbb{R}-\{1\}$. A function $f:A\rightarrow B$ is defined by $f(x)=(\frac{x-2}{x-3})$. Find whether f is one-one and onto.
Prev Next

AI Tutor Explanation

Powered by Gemini

Detailed Solution

Step 1: Check for One-One (Injective)

To check if f is one-one, we assume f(x1) = f(x2) for x1, x2 ∈ A. $$ \frac{x_1 - 2}{x_1 - 3} = \frac{x_2 - 2}{x_2 - 3} $$ Cross-multiplying gives: $$ (x_1 - 2)(x_2 - 3) = (x_2 - 2)(x_1 - 3) $$ $$ x_1x_2 - 3x_1 - 2x_2 + 6 = x_1x_2 - 3x_2 - 2x_1 + 6 $$ Simplifying the equation: $$ -3x_1 - 2x_2 = -3x_2 - 2x_1 $$ $$ x_2 = x_1 $$ Since f(x1) = f(x2) implies x1 = x2, the function is one-one.

Step 2: Check for Onto (Surjective)

To check if f is onto, we set y = f(x) and solve for x in terms of y. $$ y = \frac{x - 2}{x - 3} $$ $$ y(x - 3) = x - 2 $$ $$ xy - 3y = x - 2 $$ $$ xy - x = 3y - 2 $$ $$ x(y - 1) = 3y - 2 $$ $$ x = \frac{3y - 2}{y - 1} $$ For every y ∈ B (where y ≠ 1), there exists a corresponding x ∈ A (where x ≠ 3). Since x is defined for all y ≠ 1, the range of f is B. Thus, the function is onto.

Final Answer: The function f is both one-one and onto (bijective).

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the standard algebraic definitions of injectivity and surjectivity to a specific rational function.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a specific algorithmic process (setting f(x1)=f(x2) and solving for x in terms of y) to verify the properties of the function.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the fundamental understanding of Relations and Functions, specifically the mapping properties required for inverse functions.

More from this Chapter

MCQ_SINGLE
A function \(f:\mathbb{R}\rightarrow\mathbb{R}\) defined as \(f(x)=x^{2}-4x+5\) is:
SA
Show that the function $f:N\rightarrow N$, where N is a set of natural numbers, given by $f(n) = n-1$, if n is even, $n+1$, if n is odd, is a bijection.
MCQ_SINGLE
Let \(R_{+}\) denote the set of all non-negative real numbers. Then the function \(f:R_{+}\rightarrow R_{+}\) defined as \(f(x)=x^{2}+1\) is :
LA
Check whether the relation S in the set of real numbers R defined by $S=\{(a,b)$: where $a-b+\sqrt{2}$ is an irrational number is reflexive, symmetric or transitive.
LA
A relation R is defined on $N\times N$ (where N is the set of natural numbers) as: $(a, b)~R~(c,d)\Leftrightarrow a-c=b-d$ Show that R is an equivalence relation.
View All Questions