Question

For real $x$, let $f\left(x\right)=x^3+5x+1$, then

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

Answer 1

The correct option is B

f is one-one and onto R

Given that $f\left(x\right)=x^3+5x+1$
$\therefore f^{\prime }\left(x\right)=3x^2+5>0,\forall x\in R\Rightarrow f\left(x\right)\text{is strictly increasing on}R\Rightarrow \text{f(x) is one-one}$
Since $f\left(x\right)$ is a polynomial function which is continuous and increasing on $R$
with ${lim}_{x\to -\infty }f\left(x\right)=-\infty$
and ${lim}_{x\to \infty }f\left(x\right)=\infty \therefore$ Range of $f=(-\infty ,\infty )=R$
Hence $f$ is onto also. So, $f$ is one-one and onto $R$.