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. one-one and onto
• B. one-one and into
• C. many-one and onto
• D. many-one and into

Answer 1

The correct option is A one-one and onto

$f\left(x\right)=x^3+x^2+3x+\sin x,x\in R$
$f^{\prime }\left(x\right)=3x^2+2x+3+\cos x$
$f^{\prime }\left(x\right)=g\left(x\right)+\cos x$
$g\left(x\right)>0$ $[\because D=4-36=-32<0]$
Range of $g\left(x\right)$ is $[\frac{-D}{4a},\infty )$
$[\frac{+32}{12},\infty )=[\frac{8}{3},\infty )$

$\therefore f^{\prime }\left(x\right)>0$
Hence, function is strictly increasing.
$\underset{x\to \infty }{lim}f\left(x\right)=\infty$
and $\underset{x\to -\infty }{lim}f\left(x\right)=-\infty$
$\therefore$ Function is one-one and onto as $f\left(x\right)$ is continuous function.