For a real number x, let [x] denote the greatest integer less than or…

The correct answer is Option (3) → both one-one and onto

We have,

$f(x)=2x+k+ \sin x \cos x$ for $k≤x<k+1$,

where k is an integer.

$∴f'(x) = 2+ \cos 2x$ for $k < x <k +1$

$⇒f'(x) > 0$ for all $x ∈ (k, k + 1)$

$⇒f(x)$ is increasing on R

Hence, f is both one-one and onto