$Letthefunctionf:R\to Rbedefinedbyf\left(x\right)=2x+\sin xforx\in R,thenfis$

$\begin{array}{l}f\left(x\right)=2x+\sin x\\ f^1\left(x\right)=2+\cos x>0[\because -1\le \cos x\le 1]\\ \therefore fisIncre\sin gfunction\\ \therefore fisone-one\\ \because -1\le \sin x\le 1\\ \forall x\in R-\infty <2x+\sin x<\infty \\ \therefore Range=(-\infty ,\infty )=Codomain\\ \therefore fisonto\end{array}$