In section 4.3 of O'Neill's Elementary Differential Geometry, he said "Suppose f is a real-valued function on a surface M. If $x:D\rightarrow M$ is a coordinate patch in M, then the composite function f(x) is called a coordinate expression for f"
So far, so good. Then he went on to say, "For a function $F:R^n\rightarrow M$, each patch gives a coordinate expression $x^{-1}(F)$ for F."
This loses me. In the first statement, f is a real-valued function on the surface. In the second F is a function into the surface (the opposite direction). How does he make the transition from one to the other? What does the second statement mean?
If $x : D\to M$ and $F : R^n \to M$, then $x^{-1}F$ is a superposition which maps $R^n$ back to $D$, the coordinate patch: $x^{-1}(F) : R^n\to M\to D$, or $x^{-1}(F):R^n\to D$.
Consider an example. Let $x : [0,\pi]\times[0,2\pi]\to S^2$ be a coordinate patch on a sphere, specifying the usual coordinates $\theta$ and $\phi$. An example of a map $F$ could be a projection of a unit sphere onto a plane $z=0$ by casting rays from the northern pole, the point with $x=y=0$, $z=1$ (stereographic projection). We can then ask, what is the representation of such map in terms of the coordinate patch? In this example, $D=[0,\pi]\times[0,2\pi]$ and $x^{-1}(F)$ will be expressed by the functions $\theta(x,y)$ and $\phi(x,y)$. These functions give coordinate expression for $F$, in our case stereographic projection.