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

f is one-one but not onto R f is onto R but not one-one f is one-one and onto R f is neither one-one nor onto R

f is one-one and onto R

Given $f(x)=x^3+5x+1$ Now $f'(x)=3x^2+5>0,∀\,x∈R$ ∴ f (x) is strictly increasing function ∴ It is one-one Clearly, f (x) is a continuous function also increasing on R, $\underset{x→-∞}{\lim}f(x)=-∞$ and $\underset{x→∞}{\lim}f(x)=∞$ ∴ f (x) takes every value between -∞ and ∞ Thus, f (x) is onto function.