Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs (x, f(x)), such that x is in the domain of the function f. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in the Euclidean plane and thus form a subset of this plane, which is a curve in the case of a continuous function. This graphical representation of the function is also called the graph of the function. In mathematics, the graph of a function f is, formally, the set of all ordered pairs (x, f(x)), such that x is in the domain of the function f. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in the Euclidean plane and thus form a subset of this plane, which is a curve in the case of a continuous function. This graphical representation of the function is also called the graph of the function. In the case of functions of two variables, that is functions whose domain consists of pairs (x, y), the graph can be identified with the set of all ordered triples ((x, y, f(x, y)). For a continuous real-valued function of two real variables, the graph is a surface. The concept of the graph of a function is generalized to the graph of a relation. To test whether a graph of a relation represents a function of the first variable x, one uses the vertical line test. To test whether a graph represents a function of the second variable y, one uses the horizontal line test. If the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function in the line y = x. 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. 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 doesn't 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 f : X → Y {displaystyle f:X o Y} , in other words a function f {displaystyle f} together with its domain X {displaystyle X} and codomain Y {displaystyle Y} , the graph of the mapping is the set which is a subset of X × Y {displaystyle X imes Y} . In the abstract definition of a function, G ( f ) {displaystyle G(f)} is actually equal to f {displaystyle f} . The graph of the function f : { 1 , 2 , 3 } ↦ { a , b , c , d } {displaystyle f:{1,2,3}mapsto {a,b,c,d}} defined by is the subset of the set { 1 , 2 , 3 } × { a , b , c , d } {displaystyle {1,2,3} imes {a,b,c,d}}