Injectivity of a cubic

Text

Given $f(x)=x^{3}+2x$, show that it is injective on $\mathbb{R}$ and find $f^{-1}(3)$.

Solution

Injectivity. If $x_1<x_2$ then $x_1^{3}<x_2^{3}$ (cubing is increasing) and $2x_1<2x_2$; adding, $x_1^{3}+2x_1<x_2^{3}+2x_2$, i.e. $f(x_1)<f(x_2)$. Hence $f$ is strictly increasing, therefore injective. Value of the inverse. Seek $x$ with $x^{3}+2x=3$. Try $x=1$: $1+2=3$. Since $f$ is injective, this is the only solution: $\boxed{\,f^{-1}(3)=1\,}$