Search: bounded below

Question Hard Mathematics Relations and Functions Real valued functions, domain and range of these functions Class 11 Level 32

फलन (f(x)=3-\sqrt{2x-1}) का परिसर क्या है?

What is the range of (f(x)=3-\sqrt{2x-1})?

Explanation opens after your attempt

Correct Answer

A. ( \(-\infty,3] \)

Step 1

Concept

Since \(\sqrt{2x-1}\ge0\), \(3-\sqrt{2x-1}\le3\) and is unbounded below. The negative sign reverses the direction of the range.

Step 2

Why this answer is correct

The correct answer is A. ( \(-\infty,3] \). Since \(\sqrt{2x-1}\ge0\), \(3-\sqrt{2x-1}\le3\) and is unbounded below. The negative sign reverses the direction of the range.

Step 3

Exam Tip

\(\sqrt{2x-1}\ge0\) इसलिए \(3-\sqrt{2x-1}\le3\) और नीचे अनबाउंड है। ऋण चिह्न सीमा की दिशा बदल देता है।

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Relations and Functions
Topic: Real valued functions, domain and range of these functions
Difficulty: Hard
Level: 32

Question Hindi:
फलन (f(x)=3-\sqrt{2x-1}) का परिसर क्या है?

Question English:
What is the range of (f(x)=3-\sqrt{2x-1})?

Options:
A. ( (-\infty,3] )
B. ( [3,\infty) )
C. ( [1,\infty) )
D. ( \mathbb{R} )

Correct Answer:
A. ( (-\infty,3] )

Explanation Hindi:
(\sqrt{2x-1}\ge0) इसलिए (3-\sqrt{2x-1}\le3) और नीचे अनबाउंड है। ऋण चिह्न सीमा की दिशा बदल देता है।

Explanation English:
Since (\sqrt{2x-1}\ge0), (3-\sqrt{2x-1}\le3) and is unbounded below. The negative sign reverses the direction of the range.

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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)=x^-2+\sin x), तो नीचे दिए गए कथनों में निश्चित रूप से सही कौन सा है?

If \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x^-2+\sin x), which of the following statements is definitely correct?

Explanation opens after your attempt

Correct Answer

B. (f) सर्वाच्छादक नहीं है क्योंकि बहुत बड़ी ऋणात्मक संख्याएँ छवि नहीं बनतीं(f) is not onto because very large negative numbers are not images

Step 1

Concept

Since \(x^2\ge0\) and \(\sin x\ge-1\), (f(x)\ge-1).

Step 2

Why this answer is correct

The codomain \(\mathbb{R}\) contains values like (-2), but they cannot be images.

Step 3

Exam Tip

Even without the exact range, a lower bound can disprove onto property. चरण 1: \(x^2\ge0\) और \(\sin x\ge-1\), इसलिए (f(x)\ge-1)। चरण 2: सहप्रांत \(\mathbb{R}\) में (-2) जैसे मान हैं, पर वे छवि नहीं बन सकते। चरण 3: पूर्ण परास न मिले तब भी निचली सीमा दिखाकर सर्वाच्छादकता तोड़ी जा सकती है।

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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)=x^2+\sin x), तो नीचे दिए गए कथनों में निश्चित रूप से सही कौन सा है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}), (f(x)=x^2+\sin x), which of the following statements is definitely correct?

Options:
A. (f) सर्वाच्छादक है / (f) is onto
B. (f) सर्वाच्छादक नहीं है क्योंकि बहुत बड़ी ऋणात्मक संख्याएँ छवि नहीं बनतीं / (f) is not onto because very large negative numbers are not images
C. (f) का परास (\mathbb{R}) है / The range of (f) is (\mathbb{R})
D. (f) केवल ऋणात्मक मान देता है / (f) gives only negative values

Correct Answer:
B. (f) सर्वाच्छादक नहीं है क्योंकि बहुत बड़ी ऋणात्मक संख्याएँ छवि नहीं बनतीं / (f) is not onto because very large negative numbers are not images

Explanation Hindi:
चरण 1: (x^2\ge0) और (\sin x\ge-1), इसलिए (f(x)\ge-1)। चरण 2: सहप्रांत (\mathbb{R}) में (-2) जैसे मान हैं, पर वे छवि नहीं बन सकते। चरण 3: पूर्ण परास न मिले तब भी निचली सीमा दिखाकर सर्वाच्छादकता तोड़ी जा सकती है।

Explanation English:
Step 1: Since (x^2\ge0) and (\sin x\ge-1), (f(x)\ge-1). Step 2: The codomain (\mathbb{R}) contains values like (-2), but they cannot be images. Step 3: Even without the exact range, a lower bound can disprove onto property.

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

यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x^-6+1) है, तो सही कथन क्या है?

If \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x^-6+1), which statement is correct?

Explanation opens after your attempt

Correct Answer

A. यह आच्छादी नहीं हैIt is not onto

Step 1

Concept

Since \(x^6\ge0\), \(x^6+1\ge1\).

Step 2

Why this answer is correct

The codomain \(\mathbb{R}\) contains (0) and negative values, which are not obtained.

Step 3

Exam Tip

Even powers with a positive shift have a range bounded below. चरण 1: \(x^6\ge0\), इसलिए \(x^6+1\ge1\)। चरण 2: सहप्रांत \(\mathbb{R}\) में (0) और ऋणात्मक मान हैं, जो प्राप्त नहीं होते। चरण 3: सम घात और धनात्मक जोड़ वाले फलन का परास नीचे से सीमित होता है।

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

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}), (f(x)=x^6+1) है, तो सही कथन क्या है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}), (f(x)=x^6+1), which statement is correct?

Options:
A. यह आच्छादी नहीं है / It is not onto
B. यह आच्छादी है / It is onto
C. यह (-1) देता है / It gives (-1)
D. यह सभी वास्तविक मान देता है / It gives all real values

Correct Answer:
A. यह आच्छादी नहीं है / It is not onto

Explanation Hindi:
चरण 1: (x^6\ge0), इसलिए (x^6+1\ge1)। चरण 2: सहप्रांत (\mathbb{R}) में (0) और ऋणात्मक मान हैं, जो प्राप्त नहीं होते। चरण 3: सम घात और धनात्मक जोड़ वाले फलन का परास नीचे से सीमित होता है।

Explanation English:
Step 1: Since (x^6\ge0), (x^6+1\ge1). Step 2: The codomain (\mathbb{R}) contains (0) and negative values, which are not obtained. Step 3: Even powers with a positive shift have a range bounded below.

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

यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x^-4+x^-2) है, तो कौन सा कथन सही है?

If \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x^-4+x^-2), which statement is correct?

Explanation opens after your attempt

Correct Answer

A. यह आच्छादी नहीं हैIt is not onto

Step 1

Concept

Since \(x^4\ge0\) and \(x^2\ge0\), (f(x)\ge0).

Step 2

Why this answer is correct

The codomain \(\mathbb{R}\) contains negative numbers, which are not in the range.

Step 3

Exam Tip

Even-power polynomials are often bounded below. चरण 1: \(x^4\ge0\) और \(x^2\ge0\), इसलिए (f(x)\ge0)। चरण 2: सहप्रांत \(\mathbb{R}\) में ऋणात्मक संख्याएँ हैं, जो परास में नहीं आ सकतीं। चरण 3: सम घात वाले बहुपद में अक्सर परास नीचे से सीमित होता है।

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

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}), (f(x)=x^4+x^2) है, तो कौन सा कथन सही है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}), (f(x)=x^4+x^2), which statement is correct?

Options:
A. यह आच्छादी नहीं है / It is not onto
B. यह आच्छादी है / It is onto
C. यह हर ऋणात्मक संख्या देता है / It gives every negative number
D. यह केवल एकैकी है / It is only one-one

Correct Answer:
A. यह आच्छादी नहीं है / It is not onto

Explanation Hindi:
चरण 1: (x^4\ge0) और (x^2\ge0), इसलिए (f(x)\ge0)। चरण 2: सहप्रांत (\mathbb{R}) में ऋणात्मक संख्याएँ हैं, जो परास में नहीं आ सकतीं। चरण 3: सम घात वाले बहुपद में अक्सर परास नीचे से सीमित होता है।

Explanation English:
Step 1: Since (x^4\ge0) and (x^2\ge0), (f(x)\ge0). Step 2: The codomain (\mathbb{R}) contains negative numbers, which are not in the range. Step 3: Even-power polynomials are often bounded below.

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

यदि \(f:[0,\infty\)\to\mathbb{R}), (f(x)=x^-3-3x) है, तो (f) आच्छादी क्यों नहीं है?

If \(f:[0,\infty\)\to\mathbb{R}), (f(x)=x^-3-3x), why is (f) not onto?

Explanation opens after your attempt

Correct Answer

A. क्योंकि इसका परास नीचे से सीमित हैBecause its range is bounded below

Step 1

Concept

On the given domain, the minimum value occurs at (x=1) and is (-2).

Step 2

Why this answer is correct

Therefore, a real codomain value like (-3) cannot be obtained.

Step 3

Exam Tip

Changing the domain can change the onto nature of the same formula. चरण 1: दिए गए प्रांत पर फलन का न्यूनतम मान (x=1) पर (-2) है। चरण 2: इसलिए (-3) जैसा वास्तविक सहप्रांत मान नहीं मिल सकता। चरण 3: प्रांत बदलने से उसी सूत्र का आच्छादीपन बदल सकता है।

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

Question Hindi:
यदि (f:[0,\infty)\to\mathbb{R}), (f(x)=x^3-3x) है, तो (f) आच्छादी क्यों नहीं है?

Question English:
If (f:[0,\infty)\to\mathbb{R}), (f(x)=x^3-3x), why is (f) not onto?

Options:
A. क्योंकि इसका परास नीचे से सीमित है / Because its range is bounded below
B. क्योंकि यह केवल धनात्मक मान देता है / Because it gives only positive values
C. क्योंकि (0) नहीं मिलता / Because (0) is not obtained
D. क्योंकि यह फलन नहीं है / Because it is not a function

Correct Answer:
A. क्योंकि इसका परास नीचे से सीमित है / Because its range is bounded below

Explanation Hindi:
चरण 1: दिए गए प्रांत पर फलन का न्यूनतम मान (x=1) पर (-2) है। चरण 2: इसलिए (-3) जैसा वास्तविक सहप्रांत मान नहीं मिल सकता। चरण 3: प्रांत बदलने से उसी सूत्र का आच्छादीपन बदल सकता है।

Explanation English:
Step 1: On the given domain, the minimum value occurs at (x=1) and is (-2). Step 2: Therefore, a real codomain value like (-3) cannot be obtained. Step 3: Changing the domain can change the onto nature of the same formula.

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Search Class 10 Questions

फलन (f(x)=3-\sqrt{2x-1}) का परिसर क्या है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x-2+\sin x), तो नीचे दिए गए कथनों में निश्चित रूप से सही कौन सा है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x-6+1) है, तो सही कथन क्या है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x-4+x-2) है, तो कौन सा कथन सही है?

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यदि \(f:[0,\infty\)\to\mathbb{R}), (f(x)=x-3-3x) है, तो (f) आच्छादी क्यों नहीं है?

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यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x^-2+\sin x), तो नीचे दिए गए कथनों में निश्चित रूप से सही कौन सा है?

यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x^-6+1) है, तो सही कथन क्या है?

यदि \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=x^-4+x^-2) है, तो कौन सा कथन सही है?

यदि \(f:[0,\infty\)\to\mathbb{R}), (f(x)=x^-3-3x) है, तो (f) आच्छादी क्यों नहीं है?