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Question Expert Mathematics Relations and Functions Onto function Class 12 Level 25

यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=\lfloor x\rfloor+x), तो सर्वाच्छादकता के बारे में कौन सा कथन सही है?

If \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=\lfloor x\rfloor+x), which statement about onto property is correct?

Explanation opens after your attempt

Correct Answer

B. यह सर्वाच्छादक नहीं क्योंकि ((1,2)) के कुछ मान छूटते हैंIt is not onto because some values in ((1,2)) are missed

Step 1

Concept

If \(x\in[n,n+1\)), then \(\lfloor x\rfloor=n\) and \(f(x)\in[2n,2n+1\)).

Step 2

Why this answer is correct

Hence values in intervals such as ((1,2)) are missed.

Step 3

Exam Tip

For floor functions, split the domain into unit intervals. चरण 1: यदि \(x\in[n,n+1\)), तो \(\lfloor x\rfloor=n\) और \(f(x)\in[2n,2n+1\))। चरण 2: इसलिए ((1,2)) जैसे अंतराल के मान छूट जाते हैं। चरण 3: महत्तम पूर्णांक वाले फलनों में प्रांत को छोटे अंतरालों में बाँटें।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: Onto function
Difficulty: Expert
Level: 25

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}), (f(x)=\lfloor x\rfloor+x), तो सर्वाच्छादकता के बारे में कौन सा कथन सही है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}), (f(x)=\lfloor x\rfloor+x), which statement about onto property is correct?

Options:
A. यह सर्वाच्छादक है / It is onto
B. यह सर्वाच्छादक नहीं क्योंकि ((1,2)) के कुछ मान छूटते हैं / It is not onto because some values in ((1,2)) are missed
C. यह केवल पूर्णांक देता है / It gives only integers
D. यह केवल गैरऋणात्मक मान देता है / It gives only non-negative values

Correct Answer:
B. यह सर्वाच्छादक नहीं क्योंकि ((1,2)) के कुछ मान छूटते हैं / It is not onto because some values in ((1,2)) are missed

Explanation Hindi:
चरण 1: यदि (x\in[n,n+1)), तो (\lfloor x\rfloor=n) और (f(x)\in[2n,2n+1))। चरण 2: इसलिए ((1,2)) जैसे अंतराल के मान छूट जाते हैं। चरण 3: महत्तम पूर्णांक वाले फलनों में प्रांत को छोटे अंतरालों में बाँटें।

Explanation English:
Step 1: If (x\in[n,n+1)), then (\lfloor x\rfloor=n) and (f(x)\in[2n,2n+1)). Step 2: Hence values in intervals such as ((1,2)) are missed. Step 3: For floor functions, split the domain into unit intervals.

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यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=\lfloor x\rfloor+x), तो सर्वाच्छादकता के बारे में कौन सा कथन सही है?

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