Question

Let the function $f:[0,\infty )\to R$ be defined by $f\left(x\right)=2x+\sin x.$ Then $f$ is

• A. surjective but not injective
• B. bijective function
• C. neither injective nor surjective
• D. injective but not surjective

Answer 1

The correct option is D injective but not surjective

$f\left(x\right)=2x+\sin x$
$f^{\prime }\left(x\right)=2+\cos x>0$ for all $x\in [0,\infty )$
$\Rightarrow f$ is strictly increasing on $[0,\infty )$
$\therefore f$ is injective function.

For $y=-10,$
$2x+\sin x=-10$
$\Rightarrow 2x+10=\sin x$ which has no solution in $[0,\infty )$ as $2x+10\ge 10$
$\therefore f$ is not surjective.