Let f:R→R be a function such that f(x)=ax+3sin ⁡x+4 cos x.Then f(x) is invertible if

f′(x)=a+3cos ⁡x−4sin ⁡x        =a+5cos ⁡(x+α), where cos ⁡α=35For invertible, f′(x)≥0∀x or f′(x)≤0∀x i.e.,a+5cos⁡(x+α)≥0 or a+5cos⁡(x+α)≤0 i.e.,a≥−5cos⁡(x+α) or a≤−5cos⁡(x+α) i.e.,a≥5 or a≤−5 f (x) must be monotonic. Thus,