Question

Let $f:R\to R$ be a function defined by $f\left(x\right)=x^3+x^2+3x+\sin x.$ Then $f$ is

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

Answer 1

The correct option is A injective and surjective

$f\left(x\right)=x^3+x^2+3x+\sin x,x\in R$
$f^{\prime }\left(x\right)=3x^2+2x+3+\cos x$
$f^{\prime }\left(x\right)=g\left(x\right)+\cos x$
$g\left(x\right)>0$ as $D=4-36=-32<0$
Range of $g$ is $[\frac{-D}{4a},\infty )$
i.e., range of $g$ is $[\frac{+32}{12},\infty )=[\frac{8}{3},\infty )$
Also, $-1\le \cos x\le 1$
$\therefore f^{\prime }\left(x\right)>0$
Hence, function $f$ is strictly increasing.
$\Rightarrow f$ is injective.

$\underset{x\to \infty }{lim}f\left(x\right)=\infty$
and $\underset{x\to -\infty }{lim}f\left(x\right)=-\infty$
Also, $f$ is continuous over $R$.
$\Rightarrow$ Range of $f$ is $R$
$\therefore f$ is surjective.