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Concept
Graph of a function
Summary
In mathematics, the graph of a function is the set of ordered pairs , where In the common case where and are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane.
In the case of functions of two variables, that is functions whose domain consists of pairs the graph usually refers to the set of ordered triples where instead of the pairs as in the definition above. This set is a subset of three-dimensional space; for a continuous real-valued function of two real variables, it is a surface.
In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using rectangular axes; see Plot (graphics) for details.
A graph of a function is a special case of a relation.
In the modern foundations of mathematics, and, typically, in set theory, a function is actually equal to its graph. However, it is often useful to see functions as mappings, which consist not only of the relation between input and output, but also which set is the domain, and which set is the codomain. For example, to say that a function is onto (surjective) or not the codomain should be taken into account. The graph of a function on its own does not determine the codomain. It is common to use both terms function and graph of a function since even if considered the same object, they indicate viewing it from a different perspective.
Given a mapping in other words a function together with its domain and codomain the graph of the mapping is the set
which is a subset of . In the abstract definition of a function, is actually equal to
One can observe that, if, then the graph is a subset of (strictly speaking it is but one can embed it with the natural isomorphism).
The graph of the function defined by
is the subset of the set
From the graph, the domain is recovered as the set of first component of each pair in the graph .
Similarly, the range can be recovered as .
Official source
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