Question

Let function $f:R\to R$ be defined by $f(x)=2x+\sin x$ 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

Answer: (A)

Given that
$f(x)=2x+\sin x,x\in R$
$\Rightarrow f^{\prime }(x)=2+\cos x$
But $-1\le \cos x\le 1$
$\Rightarrow 1\le 2+\cos x\le 3$
$\Rightarrow 1\le 2+\cos x\le 3$
$\therefore f^{\prime }(x)>0,\forall x\in R$
$\Rightarrow f(x)$ is strictly increasing and hence one-one
Also as $x\to \infty ,f(x)\to \infty$ and
$x\to -\infty ,f(x)\to -\infty$
$\therefore$ Range of $f(x)=R=$ domain of $f(x)$
$\Rightarrow f(x)$ is onto.
Thus, $f(x)$ is one-one and onto.