Class CBSE Class 12 Mathematics Relations and Functions Q #1742
COMPETENCY BASED
APPLY
2 Marks 2026 AISSCE(Board Exam) VSA
Check whether $f:R-\{3\} \rightarrow R$ defined as $f(x)=\frac{x-2}{x-3}$ is onto or not.
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Detailed Solution

Step 1: Understand the definition of an onto function

A function $f: A \rightarrow B$ is onto (surjective) if for every element $y$ in the codomain $B$, there exists at least one element $x$ in the domain $A$ such that $f(x) = y$.

Step 2: Set up the equation

Let $y$ be an arbitrary element in the codomain $R$. We set $f(x) = y$:

$$y = \frac{x-2}{x-3}$$

Step 3: Solve for x in terms of y

Multiply both sides by $(x-3)$:

$$y(x-3) = x-2$$ $$xy - 3y = x - 2$$ $$xy - x = 3y - 2$$ $$x(y-1) = 3y - 2$$ $$x = \frac{3y-2}{y-1}$$

Step 4: Analyze the range

For $f$ to be onto, $x$ must exist in the domain $R - \{3\}$ for every $y \in R$. Looking at the expression $x = \frac{3y-2}{y-1}$, we see that $x$ is undefined when $y = 1$. This means there is no $x$ in the domain such that $f(x) = 1$.

Step 5: Conclusion

Since $y=1$ is in the codomain $R$ but has no pre-image in the domain, the function is not onto.

Final Answer: The function is not onto.

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the algebraic definition of surjectivity to a rational function and perform algebraic manipulation to test the existence of pre-images.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a specific algorithmic process (setting f(x)=y, solving for x, checking domain constraints) to determine the property of the function.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the conceptual understanding of Relations and Functions (Chapter 1), specifically the ability to verify mapping properties beyond rote memorization.

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