As rogerl suggests, the term "dimension" is not really used to describe a function. You are right in thinking that counting coordinates is not quite satisfactory as an approach to the notion of dimension, which is used to describe geometric figures such as the graphs of simple functions like $f(x)=2x^3$. The plane containing the coordinate axes is two dimensional in the ordinary sense of the word -- but note that "dimension" is describing a geometric object not a function. The curve defined by the function, however, is considered one dimensional even though two numbers are required to fix any point on the curve. Because giving a precise description or definition of the notion of dimension is rather difficult, you won't see much discussion of it in more elementary texts. In the case of any function graphs you are likely to be able to imagine, there are some simple intuitive, geometric properties that may convince you a function graph is quite different than an geometric object like a disk which is clearly 2 dimensional. (I use the term "disk" to refer to a circle and the entire area it encloses.) For example, if you pick a point on your function graph and erase it from the curve, you are left with two separated pieces: the curve minus the point is disconnected. (Imagine doing this with the graph of $f(x)=2x-1$ or $f(x)=4-x^2$.) Now imagine a disk, and punch out one point inside the disk. The remainder is still a single connected geometric object, not two or more separated pieces. You can't split a disk into separated pieces by removing any finite number of points, but you can do so by removing a 1 dimensional curve -- just drag your pen point right through the disk from one side to the other.
P.S. Although these disconnection properties are intuitive and capture some of what distinguishes commonly encountered 1 and 2 dimensional objects, they don't lead to a satisfactory mathematical treatment of dimension due to the existence of some pretty hairy phenomena present even in the plane, let alone in space (or higher dimensions).