Question

Let the function $f:R\to R$ be defined by $f\left(x\right)=2x+\sin x$ for $x\in R$. Then $f$ is

• A. one-one and onto
• B. one-one but not onto
• C. onto but not one-one
• D. neither one-one nor onto

Answer 1

The correct option is A one-one and onto

$f\left(x\right)=2x+\sin x\Rightarrow f^{\prime }\left(x\right)=2+\cos x\ge 1>0$ $[\because -1\le \cos x\le 1]$

$\therefore 1\le 2+\cos x\le 3$

$\Rightarrow f$ is strictly increasing on $R$

$\Rightarrow r$ is one$-$one on $R$.

Again as every vertical line does not meet the graph of $f\left(x\right)$, so $f$ onto.

Hence, $f$ is one-one and onto.