Let f : A → B be defined by f(x) = (x-2)/(x-3), where A = R - \3\ and B = R - \1\. Discuss the bijectivity of the function.

MathsCBSE Class 12 · Board exam 2025, set 65/7/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2025: question 21

Set in the CBSE Class 12 board exam of 2025, paper 65/7/1, section B, question 21, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21 · 2 marks · Very short answerRelations and Functions

Let $f : A \to B$ be defined by $f (x) = \frac{x - 2}{x - 3}$ , where $A = R - {3}$ and $B = R - {1}$ . Discuss the bijectivity of the function.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/7/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

A bijection must be both one-one and onto. Check each property separately.

Hint 2

For onto, solve $y = \frac{x - 2}{x - 3}$ for x and check that this x lies in $R - {3}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

One-one: Let $x_{1}, x_{2} \in A$ with $f (x_{1}) = f (x_{2})$ . Then $\frac{x _{1} - 2}{x _{1} - 3} = \frac{x _{2} - 2}{x _{2} - 3}$ , so $(x_{1} - 2) (x_{2} - 3) = (x_{2} - 2) (x_{1} - 3)$ . Expanding gives $- 3 x_{1} - 2 x_{2} = - 3 x_{2} - 2 x_{1}$ , i.e. $x_{1} = x_{2}$ . Hence $f$ is one-one.
1 mark
Onto: Let $y \in B = R - {1}$ and set $\frac{x - 2}{x - 3} = y$ . Then $x - 2 = x y - 3 y$ , so $x (y - 1) = 3 y - 2$ and $x = \frac{3 y - 2}{y - 1}$ , which is defined because $y \neq = 1$ . Also $x \neq = 3$ , since $\frac{3 y - 2}{y - 1} = 3$ would give $- 2 = - 3$ , which is impossible. So $x \in A$ and $f (x) = y$ . Hence $f$ is onto.
½ mark
Since $f$ is both one-one and onto, $f$ is a bijection.
½ mark

Answer: f is one-one and onto, hence bijective.

Where the marks are

Shows $f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ (1 mark)
Finds pre-image $x = \frac{3 y - 2}{y - 1} \in A$ for every $y \in B$ , so f is onto (½ mark)
Concludes f is bijective (½ mark)

Revise the chapter

Chapter: Relations and Functions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Relations and Functions, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/7/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 21

Hints

Step-by-step solution

Revise the chapter

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