Function Graph
 |  |
Given a function 
defined
on a domain 
, the graph of
is defined as the set of points (which often form
a curve or surface) showing
the values taken by 
over 
(or some portion
of 
). Technically, for real functions,
 |  |  |
(1)
|
 |  |  |
(2)
|
A graph is sometimes also called a plot. Unfortunately, the word "graph" is uniformly used by mathematicians to mean a collection of vertices and edges connecting them. In some education circles, the term "vertex-edge graph" is used in an attempt to distinguish the two types of graph. However, as Gardner (1984, p. 91) notes, "The confusion of this term with the 'graphs' of analytic geometry is regrettable, but the term has stuck [in the mathematical community]." In this work, the term "graph" will therefore be used to refer to a collection of vertices and edges, while a graph in the sense of a plot of a function will be called a "function graph" when any ambiguity arises.
Two- and three-dimensional graphs can be produced in the Wolfram Language using the commands Plot[f,
x, xmin, xmin
] and Plot3D[f,
x, xmin, xmin
, 
y, ymin,
ymax
], respectively.
Several examples of continuous functions which are notoriously difficult to graph are shown above: 
and its fractional
part. Good routines for plotting graphs use adaptive algorithms which plot more
points in regions where the function varies most rapidly (Wagon 1991, Math Works
1992, Heck 1993, Wickham-Jones 1994). Tupper (1996) has developed an algorithm that
rigorously proves the pixels it generates are "on" if and only if there
exists a mathematical point within the region of space represented by that pixel
that is a solution to the relation being graphed. Although this method attempts to
produce graphs that satisfy strict mathematical relationships, the problem of graphing
is ultimately intractable, so no fixed algorithm can produce correct graphs for arbitrary
relations.
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