Well, I have a pretty simple (probably silly and basic) doubt about how do we describe functions on a Manifold. Well, just to make clear which definitions I'm using, for now what I know is that: a topological manifold is a metric space $M$ such that for each $p\in M$ there's a integer $n$ with $n \geq 0$ such that we can find a neighborhood of $p$ homeomorphic to $\Bbb R^n$. A chart for a manifold $M$ is a pair $(x,U)$ with $U\subset M$ and $x : U \to \Bbb R^n$, if two charts overlap smoothly we say that they're $C^\infty$ related and a collection of $C^\infty$ related charts is an atlas for $M$. In that setting, a smooth manifold is a pair $(M,\mathcal{A})$ where $M$ is a topological manifold and $\mathcal{A}$ is one maximal atlas for $M$.
So if I understood, providing a chart $(x,U)$ to a manifold $M$ is a way to introduce "coordinates" on $M$. Indeed, if $I : \Bbb R^n\to \Bbb R$ is the identity function we define $x^i = I^i \circ x$ to be the "$i$-th coordinate function" on $U$. In that case, for each $p \in U$ we have the coordinates $(x^1(p), \cdots, x^n(p))$ and if I understood correctly this is a way to allow us to bring to abstract spaces the usual notions of analysis over $\Bbb R^n$ (of course, in a refined manner, so that it fits the generality of the situation).
Now the real question: this all led me to a doubt - how do we describe functions on a manifold? Well, this question might seem strange, but there's a point, if we work with $\Bbb R^n$, then each $a \in \Bbb R^n$ is an $n$-tuple of reals $a = (a^1,\dots, a^n)$ and so we can describe functions $f : \Bbb R^n \to \Bbb R$ by combining functions defined on $\Bbb R$ applied on each $a^i$. Indeed, we have the identity $I^i(a) = a^i$, and so we can easily define for instance $f : \Bbb R^2 \to \Bbb R$ by:
$$f = \sin \circ I^1 - \sqrt{(I^1)^2 - (I^2)^2}$$
Now at a point $(a,b) \in \Bbb R^2$ we would have then:
$$f(a,b)= \sin(I^1(a,b))-\sqrt{(I^1(a,b))^2-(I^2(a,b))^2}=\sin a-\sqrt{a^2+b^2}$$
So, to define the function $f(x,y)=\sin x - \sqrt{x^2+y^2}$ we can simply combine functions from $\Bbb R$ to $\Bbb R$ using the component functions of the identity function. My understanding is: we know how to build functions from $\Bbb R^n$ to $\Bbb R$ because we implicitly construct these combinations of functions constructed on $\Bbb R$ and the components of $I$.
Now, on an arbitrary manifold, points are not necessarily $n$-tuples of numbers. The projective plane for instance is the set of all lines through the origin of $\Bbb R^n$, so although charts provide $n$-tuples for each point, each point is not itself one $n$-tuple, so the construction of functions from one arbitrary manifold to $\Bbb R$ would not be simply like that.
But now here comes my understanding: we simply use the coordinate functions, exactly to work as we do in $\Bbb R^n$! For instance, let's suppose that $M$ is a smooth manifold of dimension $2$ and that $(x,U)$ is a chart, in that case $x : U \to \Bbb R^2$ and we have two coordinate functions $x^1 = I^1 \circ x$ and $x^2 = I^2 \circ x$, and with this we can express any function $f : U \to \Bbb R^2$ simply by combining $x^1$ and $x^2$ with usual functions defined on $\Bbb R$! For instance:
$$f=\sin \circ x^1 - \cos \circ x^2$$
So that given $p \in U$ we would have $f(p) = \sin(x^1(p)) - \cos(x^2(p))$ which we would know how to compute.
So is all of this correct? Is really this way that we usually set up functions on arbitrary manifolds? Is this the correct way to use charts on a manifold to work with it?
Thanks very much in advance!
The construction you give does indeed define a function on the subset $U\subset M$. In order to define a real-valued function on all of $M$ using charts, you would have to choose an atlas $\{(x_i,U_i): i\in I\}$ for $M$, and use the above construction to define functions $f_i:U_i\to\mathbb{R}$, but you would have to additionally make sure that these functions are compatible in the sense that whenever $U_i\cap U_j$ is nonempty, $f_i=f_j$ on $U_i\cap U_j$.