Prove that if $X$ is a $n$-dimensional abstract Manifold and $x\in X$ is a point, then there is an open set $0 \in U \subseteq \mathbb{R}^n$ and a parameterization $\phi: U \to V$ such that $\phi (0) = x$.
The way I think about composition functions? Am I in the right direction?
Because $X$ is a manifold, there is a parametrization $\psi : U' \subseteq \mathbb{R}^n \to V$ where $V$ is some neighbourhood of $x$. If $\psi(q) = x$ (such a point $q \in U'$ is uniquely defined since $\psi$ is supposed to be a homeomorphism), then you can shift $q$ to the origin by using (for example) a translation.
Translations are clearly homeomorphisms, so the composition is still a local parametrization for your manifold.