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

फलन \(f:\mathbb{R}\to[0,\infty\)), (f(x)=\(e^x-1\)²), सर्वाच्छादक है या नहीं?

Is \(f:\mathbb{R}\to[0,\infty\)), (f(x)=\(e^x-1\)²), onto or not?

Explanation opens after your attempt

Correct Answer

A. सर्वाच्छादक हैIt is onto

Step 1

Concept

(\(e^x-1\)²\ge0), and at (x=0), the value is (0).

Step 2

Why this answer is correct

For any \(y\ge0\), set \(e^x-1=\sqrt{y}\), giving (x=\ln\(1+\sqrt{y}\)), which is real.

Step 3

Exam Tip

For exponential expressions, logarithms help construct preimages. चरण 1: (\(e^x-1\)²\ge0) और (x=0) पर (0) मिलता है। चरण 2: किसी भी \(y\ge0\) के लिए \(e^x-1=\sqrt{y}\) लेने पर (x=\ln\(1+\sqrt{y}\)) वास्तविक मिलता है। चरण 3: घातीय फलन में धनात्मक पूर्वप्रतिबिंब बनाने के लिए लघुगणक का प्रयोग करें।

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

Question Hindi:
फलन (f:\mathbb{R}\to[0,\infty)), (f(x)=(e^x-1)^2), सर्वाच्छादक है या नहीं?

Question English:
Is (f:\mathbb{R}\to[0,\infty)), (f(x)=(e^x-1)^2), onto or not?

Options:
A. सर्वाच्छादक है / It is onto
B. सर्वाच्छादक नहीं क्योंकि (0) नहीं मिलता / It is not onto because (0) is not obtained
C. सर्वाच्छादक नहीं क्योंकि (1) नहीं मिलता / It is not onto because (1) is not obtained
D. सर्वाच्छादक नहीं क्योंकि बड़े मान नहीं मिलते / It is not onto because large values are not obtained

Correct Answer:
A. सर्वाच्छादक है / It is onto

Explanation Hindi:
चरण 1: ((e^x-1)^2\ge0) और (x=0) पर (0) मिलता है। चरण 2: किसी भी (y\ge0) के लिए (e^x-1=\sqrt{y}) लेने पर (x=\ln(1+\sqrt{y})) वास्तविक मिलता है। चरण 3: घातीय फलन में धनात्मक पूर्वप्रतिबिंब बनाने के लिए लघुगणक का प्रयोग करें।

Explanation English:
Step 1: ((e^x-1)^2\ge0), and at (x=0), the value is (0). Step 2: For any (y\ge0), set (e^x-1=\sqrt{y}), giving (x=\ln(1+\sqrt{y})), which is real. Step 3: For exponential expressions, logarithms help construct preimages.

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फलन \(f:\mathbb{R}\to[0,\infty\)), (f(x)=\(e^x-1\)2), सर्वाच्छादक है या नहीं?

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फलन \(f:\mathbb{R}\to[0,\infty\)), (f(x)=\(e^x-1\)²), सर्वाच्छादक है या नहीं?