Class CBSE Class 12 Mathematics Relations and Functions Q #1328
KNOWLEDGE BASED
UNDERSTAND
5 Marks 2024 AISSCE(Board Exam) LA
Let $A=R-\{5\}$ and $B=R-\{1\}$. Consider the function $f:A\rightarrow B$, defined by $f(x)=\frac{x-3}{x-5}$ Show that f is one-one and onto.
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Detailed Solution

Step 1: Prove f(x) is one-one

To prove that $f(x)$ is one-one, we need to show that if $f(x_1) = f(x_2)$, then $x_1 = x_2$. Let $x_1, x_2 \in A$ such that $f(x_1) = f(x_2)$. Then, $\frac{x_1 - 3}{x_1 - 5} = \frac{x_2 - 3}{x_2 - 5}$.

Step 2: Cross-multiply and simplify

Cross-multiplying, we get: $(x_1 - 3)(x_2 - 5) = (x_2 - 3)(x_1 - 5)$ $x_1x_2 - 5x_1 - 3x_2 + 15 = x_1x_2 - 5x_2 - 3x_1 + 15$ $-5x_1 - 3x_2 = -5x_2 - 3x_1$ $2x_2 = 2x_1$ $x_1 = x_2$ Since $f(x_1) = f(x_2)$ implies $x_1 = x_2$, the function $f(x)$ is one-one.

Step 3: Prove f(x) is onto

To prove that $f(x)$ is onto, we need to show that for every $y \in B$, there exists an $x \in A$ such that $f(x) = y$. Let $y \in B$, so $y \in R - \{1\}$. We want to find $x \in A$ such that $f(x) = y$, i.e., $\frac{x - 3}{x - 5} = y$.

Step 4: Solve for x in terms of y

Solving for $x$: $x - 3 = y(x - 5)$ $x - 3 = yx - 5y$ $x - yx = 3 - 5y$ $x(1 - y) = 3 - 5y$ $x = \frac{3 - 5y}{1 - y}$

Step 5: Verify x belongs to A

We need to check if $x \in A$, i.e., $x \in R - \{5\}$. If $x = 5$, then $\frac{3 - 5y}{1 - y} = 5$ $3 - 5y = 5 - 5y$ $3 = 5$, which is a contradiction. Thus, $x \neq 5$, and $x \in A$.

Step 6: Conclude f(x) is onto

Since for every $y \in B$, there exists an $x = \frac{3 - 5y}{1 - y} \in A$ such that $f(x) = y$, the function $f(x)$ is onto.

Step 7: Final Conclusion

Since $f(x)$ is both one-one and onto, it is a bijective function.

Final Answer: f is one-one and onto

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to demonstrate understanding of the definitions of one-one and onto functions and apply them to the given function.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of one-one and onto functions, rather than just recalling facts or applying a specific algorithm. It involves manipulating the function and proving properties based on definitions.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of functions, types of functions (one-one, onto), and their properties, which are core concepts in the Relations and Functions chapter.

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