Graph of a function
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.
A plot is a graphical technique for representing a data set, usually as a graph showing the relationship between two or more variables. The plot can be drawn by hand or by a mechanical or electronic plotter. Graphs are a visual representation of the relationship between variables, very useful for humans who can quickly derive an understanding which would not come from lists of values. Graphs can also be used to read off the value of an unknown variable plotted as a function of a known one. Graphs of functions are used in mathematics, sciences, engineering, technology, finance, and other areas.
In the modern foundation of mathematics known as set theory, a function and its graph are essentially the same thing.
In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)). If the function input x is a scalar, the graph is a two-dimensional graph, and for a continuous function is a curve. If the function input x is an ordered pair (x1, x2) of real numbers, the graph is the collection of all ordered triples (x1, x2, f(x1, x2)), and for a continuous function is a surface.
Informally, if x is a real number and f is a real function, graph may mean the graphical representation of this collection, in the form of a line chart: a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching. The graph of a function on real numbers may be mapped directly to the graphic representation of the function. For general functions, a graphic representation cannot necessarily be found and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis.
The concept of the graph of a function is generalized to the graph of a relation. Note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers but it is not if its codomain is the complex field.
To test whether a graph of a curve is a function of x, one uses the vertical line test. To test whether a graph of a curve is a function of 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 over the line y = x.
Source From Wikipedia