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

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.