JEE Exam > JEE Questions > (b) Let function f : R R be defined by f(x) ...
Start Learning for Free
(b) Let function f : R → R be defined by f(x) = 2x + sinx for x ∈ R. Then f is
- a)one to one and onto
- b)one to one but NOT onto
- c)onto but NOT one to one
- d)neither one to one nor onto
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
(b) Let function f : R R be defined by f(x) = 2x + sinx for x R. The...
Free Test
FREE
| Start Free Test |
Community Answer
(b) Let function f : R R be defined by f(x) = 2x + sinx for x R. The...
Solution:
One-to-one:
To prove that f is one-to-one, let's assume that there exist two distinct elements a and b in R such that f(a) = f(b). Then we have:
2a sin(a) = 2b sin(b)
Dividing both sides by 2 and using the identity sin(a) = sin(b) if and only if a = b + 2nπ or a = π - b + 2nπ, we get:
a = b + 2nπ or a = π - b + 2nπ
where n is any integer. Since a and b are distinct, we cannot have a = b + 2nπ. Therefore, we must have a = π - b + 2nπ. Substituting this value of a in terms of b in f(a) = f(b), we get:
2(π - b + 2nπ) sin(π - b + 2nπ) = 2b sin(b)
Using the identity sin(π - x) = sin(x) and simplifying, we get:
2b sin(b) = 2b sin(b)
which is true. Therefore, we have shown that if f(a) = f(b), then a = π - b + 2nπ and hence a = b. Thus, f is one-to-one.
Onto:
To prove that f is onto, let y be any element in R. We need to show that there exists an element x in R such that f(x) = y. Let's consider the equation f(x) = y and try to solve for x. We have:
2x sin(x) = y
Dividing both sides by 2 and rearranging, we get:
x = arcsin(y/2)
Note that arcsin(y/2) is defined for all y in R since the range of the sine function is [-1, 1]. Therefore, we have shown that for any y in R, there exists an x in R such that f(x) = y. Thus, f is onto.
Conclusion:
Since f is both one-to-one and onto, we can conclude that f is a bijective function.
One-to-one:
To prove that f is one-to-one, let's assume that there exist two distinct elements a and b in R such that f(a) = f(b). Then we have:
2a sin(a) = 2b sin(b)
Dividing both sides by 2 and using the identity sin(a) = sin(b) if and only if a = b + 2nπ or a = π - b + 2nπ, we get:
a = b + 2nπ or a = π - b + 2nπ
where n is any integer. Since a and b are distinct, we cannot have a = b + 2nπ. Therefore, we must have a = π - b + 2nπ. Substituting this value of a in terms of b in f(a) = f(b), we get:
2(π - b + 2nπ) sin(π - b + 2nπ) = 2b sin(b)
Using the identity sin(π - x) = sin(x) and simplifying, we get:
2b sin(b) = 2b sin(b)
which is true. Therefore, we have shown that if f(a) = f(b), then a = π - b + 2nπ and hence a = b. Thus, f is one-to-one.
Onto:
To prove that f is onto, let y be any element in R. We need to show that there exists an element x in R such that f(x) = y. Let's consider the equation f(x) = y and try to solve for x. We have:
2x sin(x) = y
Dividing both sides by 2 and rearranging, we get:
x = arcsin(y/2)
Note that arcsin(y/2) is defined for all y in R since the range of the sine function is [-1, 1]. Therefore, we have shown that for any y in R, there exists an x in R such that f(x) = y. Thus, f is onto.
Conclusion:
Since f is both one-to-one and onto, we can conclude that f is a bijective function.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
(b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer?
Question Description
(b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. 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 (b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for (b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer?.
(b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. 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 (b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for (b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer?.
Solutions for (b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of (b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
(b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for (b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of (b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice (b) Let function f : R R be defined by f(x) = 2x + sinx for x R. Then f isa)one to one and ontob)one to one but NOT ontoc)onto but NOT one to oned)neither one to one nor ontoCorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup