In John Lee's introduction to smooth manifolds a chart $(U,\varphi)$ centered at a point $p$ of a smooth manifolds $M^n$ is defined by its coordinate functions, denoted as $\varphi(p)=(x^1(p),x^2(p),\ldots,x^n(p))$. Given a smooth function $f:M\to\mathbb{R}$ its coordinate representation is $f\circ\varphi^{-1}:\varphi(U)\subseteq\mathbb{R}^n\to\mathbb{R}$, but sometimes this is refered to as $f(x^1,x^2,\ldots,x^n)$ why? In this case $(x^1,x^2,\ldots, x^n)$ is a point in $\mathbb{R}^n$?
Furthermore, my confusion comes in greater detail when we consider smooth maps between manifolds, for example, given a smooth map $f:M^n\to N^m$ its coordinate representation is of the form $\psi\circ f\circ\varphi^{-1}:\varphi(U)\to\psi(V)$, where $(U,\varphi)$ and $(V,\psi)$ are smooth charts centered at the points $p\in M$ and $f(p)\in N$, respectively. But in some proofs this isnt written, instead they say something like: $f(x^1,x^2,\ldots,x^n)=(x^{k+1},\ldots,x^n)$, such as in the following proof:
Any help is appreciated!