Show that the function f : R → R defined by f(x) = 4x³ - 5, x R is one-one and onto.

MathsCBSE Class 12 · Board exam 2025, set 65/7/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2025: question 26(a)

Set in the CBSE Class 12 board exam of 2025, paper 65/7/1, section C, question 26(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(a) · 3 marks · Short answerRelations and Functions

Show that the function $f : R \to R$ defined by $f (x) = 4 x^{3} - 5$ , $\forall x \in R$ is one-one and onto.

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

For one-one, start from $f (x_{1}) = f (x_{2})$ and simplify.

Hint 2

For onto, solve $4 x^{3} - 5 = y$ for x and check that x is real for every real 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 $x_{1}, x_{2} \in R$ with $f (x_{1}) = f (x_{2})$ . Then $4 x_{1}^{3} - 5 = 4 x_{2}^{3} - 5$ , so $x_{1}^{3} = x_{2}^{3}$ , which gives $x_{1} = x_{2}$ (the real cube root is unique). Hence $f$ is one-one.
1½ marks
Onto: Let $y \in R$ (co-domain). Take $x = (\frac{y + 5}{4})^{1/3}$ , which is real for every real y. Then $f (x) = 4 \cdot \frac{y + 5}{4} - 5 = y$ . So every element of the co-domain has a pre-image, and $f$ is onto. Hence $f$ is both one-one and onto.
1½ marks

Answer: f is one-one and onto (bijective).

Where the marks are

Proves $f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ (1½ marks)
Shows every real y has the real pre-image $3 (y + 5) /4$ , so range = co-domain (1½ marks)

Revise the chapter

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

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 26(a)

Hints

Step-by-step solution

Revise the chapter

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