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Question Medium Mathematics Relations and Functions Onto function Class 12 Level 26

यदि \(f:\mathbb{R}\to[0,1]\), (f(x)=\frac{x^-2}{1+x^-2}) है, तो (f) आच्छादी नहीं है क्योंकि

If \(f:\mathbb{R}\to[0,1]\), (f(x)=\frac{x^-2}{1+x^-2}), then (f) is not onto because

Explanation opens after your attempt

Correct Answer

A. (1) प्रतिबिंब नहीं बनता(1) is not an image

Step 1

Concept

\(\frac{x^2}{1+x^2}\) is always at least (0) and less than (1).

Step 2

Why this answer is correct

At (x=0), we get (0), but (1) is not obtained for any real (x). So the range is ([0,1)), not the codomain ([0,1]).

Step 3

Exam Tip

Distinguish between a limiting value and an attained value. चरण 1: \(\frac{x^2}{1+x^2}\) हमेशा (0) या उससे अधिक और (1) से कम होता है। चरण 2: (x=0) पर (0) मिलता है, पर (1) किसी भी वास्तविक (x) से नहीं मिलता। इसलिए परास ([0,1)) है, सहप्रांत ([0,1]) नहीं। चरण 3: सीमा मान और प्राप्त मान में अंतर रखें।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: Onto function
Difficulty: Medium
Level: 26

Question Hindi:
यदि (f:\mathbb{R}\to[0,1]), (f(x)=\frac{x^2}{1+x^2}) है, तो (f) आच्छादी नहीं है क्योंकि

Question English:
If (f:\mathbb{R}\to[0,1]), (f(x)=\frac{x^2}{1+x^2}), then (f) is not onto because

Options:
A. (1) प्रतिबिंब नहीं बनता / (1) is not an image
B. (0) प्रतिबिंब नहीं बनता / (0) is not an image
C. (\frac{1}{2}) प्रतिबिंब नहीं बनता / (\frac{1}{2}) is not an image
D. परास ([0,1]) है / The range is ([0,1])

Correct Answer:
A. (1) प्रतिबिंब नहीं बनता / (1) is not an image

Explanation Hindi:
चरण 1: (\frac{x^2}{1+x^2}) हमेशा (0) या उससे अधिक और (1) से कम होता है। चरण 2: (x=0) पर (0) मिलता है, पर (1) किसी भी वास्तविक (x) से नहीं मिलता। इसलिए परास ([0,1)) है, सहप्रांत ([0,1]) नहीं। चरण 3: सीमा मान और प्राप्त मान में अंतर रखें।

Explanation English:
Step 1: (\frac{x^2}{1+x^2}) is always at least (0) and less than (1). Step 2: At (x=0), we get (0), but (1) is not obtained for any real (x). So the range is ([0,1)), not the codomain ([0,1]). Step 3: Distinguish between a limiting value and an attained value.

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यदि \(f:\mathbb{R}\to[0,1]\), (f(x)=\frac{x-2}{1+x-2}) है, तो (f) आच्छादी नहीं है क्योंकि

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यदि \(f:\mathbb{R}\to[0,1]\), (f(x)=\frac{x^-2}{1+x^-2}) है, तो (f) आच्छादी नहीं है क्योंकि