There's a naive exercise that I'm having trouble to finish.
Suppose $M$ is a smooth manifold and $f:$ $M$ $\rightarrow$ $\mathbb{R}^{k}$ is a smooth function. Show that: $f\circ\varphi^{-1}:\varphi(U)\rightarrow\mathbb{R}^{k}$ is smooth for every smooth chart $(U,\varphi)$ for $M$.
Well, by definition of smooth function, for every point $p$ in M there is a smooth chart $(U,\varphi)$ such that the function above is smooth.
My idea is to take, for every $p$ $\in$ $M$, such a chart. Sounds like a good start, but I couldn't find an argument to finish the problem, even because there are charts $(U',\psi)$ such that $p$ $\in$ $U$ $\subset$ $U'$ that I haven't considered in the beginning.
If $U'$ $\subset$ $U$, we just have to restrict the domain, but in the first case I couldn't go on. What am I missing?