For a real number x let [x] denoutes the greatest interger less than or equal to x, let `f : R to R` be defined by `f (x) =2x+ [x]+ sin cos x,` then f is :

Options

A. one-one but not onto
B. onto but not one-one
C. both one-one and onto
D. neither one-one nor onto

Answer

C

Explanation

We have
`f(x)=2x+k+sin x cos x" for "k le x lt k+1`
Where k is an integer.
`therefore f'(x)=2+cos2x" for "k lt x lt k+1`
`Rightarrowf'(x)gt 0" for all " x in (k, k+1)`
`Rightarrow f(x)` is increasing on R.
Hence, f is both one-one and onto.