Let a differentiable manifold $M_{1}$ be the set of real numbers $\mathbb{R}$, with the coordinate $\phi_{1}(x)=x$, and $M_{2}$ also $\mathbb{R}$ but with the coordinate $\phi_{2}(x)=x^{3}$. Show that $M_{1}$ and $M_{2}$ are diffeomorphic. Find a suitable map $f: M_{1} \rightarrow M_{2}$ and show that it is a diffeomorphism.