Why is the polynomial $f(x)=x^3+x^2+x+1$ monotonic? I have to argue why the polynomial $f(x)=x^3+x^2+x+1$ has a reverse function $f^{-1}$ which is defined in on the whole of $\mathbb R$. I'm certain the argument would simply be that because $f(x)$ is monotonic on $\mathbb R$ it is also injective on $\mathbb R$. However I can not argue why $f(x)$ would be monotonic. A nudge in the right direction would be appreciated.
English is not my first language, please do say if my terminology is off.

Without calculus, you could look at $f(y)-f(x)=(y-x)(x^2+xy+y^2+x+y+1)$, and show that the right-hand side is positive whenever $y>x$.
