$f: R→ R$ given by $f(x) = 2x + |\cos x|$, is

one-one and into one-one and onto many-one and into many-one and onto

one-one and onto

Clearly, $g: R→ R$ given by $g(x) = 2x$ is a bijection. Also, it is strictly increasing function on R. Therefore, $f(x)=2x+ |\cos x|$ is also strictly increasing function on R. Hence, it is one-one and onto function.