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Let the function f: RvecR be defined by ...

Let the function `f: RvecR` be defined by `f(x)=x+sinxforx in Rdot` Then `f` is one-to-one and onto one-to-one but not onto onto but not-one-to-one neither one-to-one nor onto

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

Text Solution

Verified by Experts

The correct Answer is:
A
Given, `f(x)=2x+sinx`
`rArr f'(x)=2+cosx rArr f'(x) gt 0, AA x in R`
which shows f(x) is one-one, as f(x) is strictly increasing.
Since f(x) is increasing for every ` x in R.`
`thereforef(x)` takes all intermediate values between `(-oo,oo)`.
Range of `f(x) in R`
Hence, f(x) is one-to-one and onto.
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