A function f : R - \3/5 \ R - \3/5 \ is defined as f(x) = (3x+2)/(5x-3). Show that f is one-one and onto.

f is one-one and onto (pre-image x=(3y+2)/(5y-3))

MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2026: question 32(b)

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section D, question 32(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(b) · 5 marks · Long answerRelations and Functions

A function $f : R - {\frac{3}{5}} ⟶ R - {\frac{3}{5}}$ is defined as $f (x) = \frac{3 x + 2}{5 x - 3}$ . Show that f is one-one and onto.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/1/1.

Hints

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

Hint 1

For one-one, assume $f (x_{1}) = f (x_{2})$ and cross-multiply.

Hint 2

For onto, solve $y = \frac{3 x + 2}{5 x - 3}$ for $x$ .

Hint 3

Verify the $x$ you get is not $\frac{3}{5}$ and that $f (x) = y$ .

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 $x_{1}, x_{2} \in R - {\frac{3}{5}}$ with $f (x_{1}) = f (x_{2})$ :
$\frac{3 x _{1} + 2}{5 x _{1} - 3} = \frac{3 x _{2} + 2}{5 x _{2} - 3}$
$\Rightarrow 15 x_{1} x_{2} - 9 x_{1} + 10 x_{2} - 6 = 15 x_{1} x_{2} + 10 x_{1} - 9 x_{2} - 6$
$\Rightarrow 19 x_{2} = 19 x_{1} \Rightarrow x_{1} = x_{2}$ . Hence $f$ is one-one.
2½ marks
Onto: Let $y \in R - {\frac{3}{5}}$ (codomain). Setting $\frac{3 x + 2}{5 x - 3} = y$ : $5 x y - 3 y = 3 x + 2 \Rightarrow x (5 y - 3) = 3 y + 2 \Rightarrow x = \frac{3 y + 2}{5 y - 3}$ , defined as $y \neq = \frac{3}{5}$ . Also $x \neq = \frac{3}{5}$ , since $\frac{3 y + 2}{5 y - 3} = \frac{3}{5}$ gives $15 y + 10 = 15 y - 9$ , impossible. So $x$ is in the domain, and
$f (x) = \frac{3 ( 3 y + 2 ) + 2 ( 5 y - 3 )}{5 ( 3 y + 2 ) - 3 ( 5 y - 3 )} = \frac{19 y}{19} = y$ .
Every element of the codomain has a pre-image, so $f$ is onto.
2½ marks

Answer: f is one-one and onto (pre-image $x = \frac{3 y + 2}{5 y - 3}$ )

Where the marks are

Shows $f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ (2½ marks)
Finds $x = \frac{3 y + 2}{5 y - 3}$ in the domain with $f (x) = y$ , concluding onto (2½ marks)

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 paper2026 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 32(b)

Hints

Step-by-step solution

Revise the chapter

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