Question

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

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

Answer 1

Answer: (C)

$f$ is one-one and onto

$f\left(x\right)=x^3+5x+1f\left(x\right)\in (-\infty ,\infty )f^{{}^{\prime }}\left(x\right)={3x}^2+5$

$f^{{}^{\prime }}\left(x\right)>0\forall x\in R$

$\therefore f\left(x\right)$ is always an increasing function so $f\left(x\right)$ is one-one and onto.