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

$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3+5\mathrm{x}+1,$ for all $\mathrm{x}\in \mathrm{R}$
$\mathrm{f}\text{'}\left(\mathrm{x}\right)=3{\mathrm{x}}^2+5>0$
$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)$is montonic on R.
Also f(x) being polynomial of degree and so range of f(x) is R.
$\Rightarrow$f is one-one and onto R.