A function f:R→R defined as f(x)=x²-4x+5 is : (A) injective but not surjective. (B) surjective but not injective. (C) both injective and surjective. (D) neither injective nor surjective.

MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 1 mark · MCQ

CBSE Class 12 Mathematics 2024: question 1

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, section A, question 1, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q1 · 1 mark · MCQRelations and Functions

A function $f : R \to R$ defined as $f (x) = x^{2} - 4 x + 5$ is :

(A)injective but not surjective.
(B)surjective but not injective.
(C)both injective and surjective.
(D)neither injective nor surjective.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

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

Hint 1

Complete the square in $x^{2} - 4 x + 5$ .

Hint 2

Compare $f (1)$ and $f (3)$ , and find the least value of $f$ .

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.

$f (x) = (x - 2)^{2} + 1$ . Since $f (1) = f (3) = 2$ with $1 \neq = 3$ , $f$ is not one-one.
½ mark
Also $f (x) \geq 1$ for all $x$ , so the range is $[1, \infty) \neq = R$ ; for example, $0$ has no pre-image. Hence $f$ is not onto, and the answer is option (D).
½ mark

Answer: (D) neither injective nor surjective

Where the marks are

Shows $f$ is not one-one and its range is $[1, \infty)$ , choosing (D) (1 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 paper2024 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 1

Hints

Step-by-step solution

Revise the chapter

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