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If f : R → S, defined by f(x) = sin x – √3 cos x+ 1, is onto, then the interval of S is-
[AIEEE 2004]
- a)[0, 3]
- b)[–1, 1]
- c)[0, 1]
- d)[–1, 3]
Correct answer is option 'D'. Can you explain this answer?
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If f : R S, defined by f(x) = sin x 3 cos x+ 1, is onto, then the in...
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If f : R S, defined by f(x) = sin x 3 cos x+ 1, is onto, then the in...
Understanding the Function f(x)
The function given is f(x) = sin(x) + 3cos(x) + 1. To determine if f is onto, we need to find the range of this function.
Step 1: Analyze the Components
- The function consists of a sine component, sin(x), which varies between -1 and 1.
- The cosine component, 3cos(x), varies between -3 and 3.
Step 2: Combine the Components
- Adding these components together, we get:
f(x) = sin(x) + 3cos(x) + 1
- The minimum value occurs when sin(x) is -1 and cos(x) is -1:
Minimum = -1 - 3 + 1 = -3
- The maximum value occurs when sin(x) is 1 and cos(x) is 1:
Maximum = 1 + 3 + 1 = 5
Step 3: Determine Range
- Thus, we have established that the range of f(x) is from -3 to 5.
Step 4: Finding the Interval when f is Onto
- For f to be onto, it must map to every value within a certain interval in S.
- The function can achieve values continuously between its minimum and maximum due to the periodic nature of sine and cosine functions.
Conclusion: The Interval of S
- Considering the options:
- a) [0, 3]
- b) [1, 1]
- c) [0, 1]
- d) [1, 3]
- The correct choice is option d) [1, 3] since the function f(x) can indeed achieve all values between 1 and 3 inclusively based on the derived range from -3 to 5.
This indicates that the function is onto within the specified interval S.
The function given is f(x) = sin(x) + 3cos(x) + 1. To determine if f is onto, we need to find the range of this function.
Step 1: Analyze the Components
- The function consists of a sine component, sin(x), which varies between -1 and 1.
- The cosine component, 3cos(x), varies between -3 and 3.
Step 2: Combine the Components
- Adding these components together, we get:
f(x) = sin(x) + 3cos(x) + 1
- The minimum value occurs when sin(x) is -1 and cos(x) is -1:
Minimum = -1 - 3 + 1 = -3
- The maximum value occurs when sin(x) is 1 and cos(x) is 1:
Maximum = 1 + 3 + 1 = 5
Step 3: Determine Range
- Thus, we have established that the range of f(x) is from -3 to 5.
Step 4: Finding the Interval when f is Onto
- For f to be onto, it must map to every value within a certain interval in S.
- The function can achieve values continuously between its minimum and maximum due to the periodic nature of sine and cosine functions.
Conclusion: The Interval of S
- Considering the options:
- a) [0, 3]
- b) [1, 1]
- c) [0, 1]
- d) [1, 3]
- The correct choice is option d) [1, 3] since the function f(x) can indeed achieve all values between 1 and 3 inclusively based on the derived range from -3 to 5.
This indicates that the function is onto within the specified interval S.
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If f : R S, defined by f(x) = sin x 3 cos x+ 1, is onto, then the interval of S is-[AIEEE 2004]a)[0, 3]b)[1, 1]c)[0, 1]d)[1, 3]Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If f : R S, defined by f(x) = sin x 3 cos x+ 1, is onto, then the interval of S is-[AIEEE 2004]a)[0, 3]b)[1, 1]c)[0, 1]d)[1, 3]Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If f : R S, defined by f(x) = sin x 3 cos x+ 1, is onto, then the interval of S is-[AIEEE 2004]a)[0, 3]b)[1, 1]c)[0, 1]d)[1, 3]Correct answer is option 'D'. Can you explain this answer?.
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