Let  $\text{f:}R\to R$  be a function such that $f\left(x\right)=ax+3 s\mathrm{in}x+4\cos x$ .Then $f\left(x\right)$  is invertible if

$f^{\mathrm{\prime }}\left(x\right)=a+3\cos x-4\sin x$

$=a+5\cos (x+\alpha )$,  where  $\cos \alpha =\frac{3}{5}$

For invertible, $f\left(x\right)$  must be monotonic. Thus,

$f^{\mathrm{\prime }}\left(x\right)\ge 0\mathrm{\forall }x \text{or }{f}^{\mathrm{\prime }}\left(x\right)\le 0\mathrm{\forall }x$

i.e.,  $a+5\cos (x+\alpha )\ge 0 \text{or }a+5\cos (x+\alpha )\le 0$

i.e.,   $a\ge -5\cos (x+\alpha ) \text{or }a\le -5\cos (x+\alpha )$

i.e., $a\ge 5 \text{or }a\le -5$