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Why is a function $f(x)$ called a single-variable function if it has coordinates represented by $x$ and $y$? Can it be called a 1D function if its plot is 2D? Subsequently, can two-variable functions $f(x,y)$ (that have coordinates $x,y,z$) be called a 2D function if its plot is 3D?

This is confusing and potentially ambiguous. How should one refer to these types of functions in order to discriminate them clearly?

Please provide sources if possible.

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  • $\begingroup$ The graph of a smooth function of n variables is n dimensional. $\endgroup$ – mathematician Sep 4 '14 at 0:19
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    $\begingroup$ $f(x)$ is a one-variable function; I've not heard it called a one-dimensional function. Its graph is a line - something that looks like a one-dimensional subset of a two-dimensional space. I'm not sure exactly what your confusion is. $\endgroup$ – rogerl Sep 4 '14 at 0:19
  • $\begingroup$ @rogerl If the graph is a curve, then would it not be a two-dimensional subset of two-dimensional space? In any case, I am looking for clarification on why a function with coordinates (x,y) is called a single-variable function. $\endgroup$ – Daniel W Sep 4 '14 at 0:27
  • $\begingroup$ No; if you "pulled" on both ends of the graph, it would be a straight line. If you look at a small portion of the graph, it looks like a line. The graph is one-dimensional; it happens to be embedded in a two-dimensional space. $\endgroup$ – rogerl Sep 4 '14 at 0:36
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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).

HTH some.

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