If f:R→R given by f(x)=ax+sin⁡x+a, then f is one-one and onto for all

For a≠0, the range of f is R. f is differentiable function so f is one-one and only if f is monotonic.    f′(x)=a+cos⁡xIf a>1, f'(x)>0 i.e. f is increasingIf a<-1, f'(x)<0 i.e. f is decreasing Thus f is monotonic if a∈R~[-1,1].