Graphs, Domains, and Ranges | Research & Encyclopedia Articles

Graphs, Domains, and Ranges

Graphs provide a means to visualize a mathematical set, function, or even set of functions. Objects depicted in a graph may be one-dimensional, such as a number line; two-dimensional, such as a parabola, or three-dimensional, such as a sphere. Graphs are used extensively throughout the fields of mathematics, statistics, science, engineering, business, and research in order to depict information--whether a mathematical curve of statistical data--in a form that humans find easy to process. Graphs provide a means to quickly assimilate information--and even trends present in the information--in a more intuitive way than reviewing long lists of numerical data.

Domain

The domain of a function, often symbolized by an italicized capital D, is the set of values over which the function exists. The set of values is generally connected (that is, continuous, such as all values in the closed interval [-1, 1]), but may include points or intervals of discontinuity, especially with piecewise functions. For example, the domain of the function f(x) = x² is negative infinity to positive infinity (that is, all real numbers), with no discontinuities. The domain of the function f(x) = 1 / x, however, is the set of all real numbers, excepting x = 0 since it is impossible to divide by zerozero. Similarly, the domain of the function f(x) = (x - 1)² is the set of all real numbers, excepting x = 1, which would also result in a division by zero error. The domain of f(x) = 1 / x is often written (-(, 0) ( (0, () which reads "the set of all real numbers in the open interval negative infinity to zero union the set of all numbers in the open interval zero to infinity." The domain of f(x) = (x - 1)² is similarly written (-(, 1) ( (1, (). It is important to note that domains may also be composed of complex numbers, if the function is defined in the complex number plane.

Range

The range of a function, often symbolized by an italicized capital R, is the set of values over which the domain of a function is mapped by the function. The mapping may be one-to-one, which means that the function maps each unique value in the domain to a unique value in the range, or the mapping may be many-to-one, which means that the function maps two or more values in the domain to the same value in the range. Written symbolically, a one-to-one function implies that if f(x) = f(y) then x = y. Many-to-one functions do not have this restriction. A function's codomain, which is sometimes confused with a function's range, is any set of values that contains the range; thus, the codomain may or may not be equal to the range. For example, if the range of a function is [-1, 1] (such as the function sin x or cos x), a possible codomain of the function is [-2, 2], which is clearly not the same as [-1, 1]. It is important to note that the domain and range of a function may be the same, but this condition is unlikely. For example, the domain of f(x) = sin x is (-(, (), but its range is [-1, 1].

The functions f(x) = 2x and f(x) = 3x³ - 2 are examples of one-to-one functions with a range of (-(, (). For both functions, each value x in the domain has a unique value f(x) in the range. The functions f(x) = |x| and f(x) = 4x², however, are many-to-one functions with a range of [0, (). When f(x) = |x|, for example, f(-1) = f(1) = 1 and when f(x) = 4x², f(-2) = f(2) = 16. For these functions, each unique value x in the domain does not map to a unique value in the range. Like the domain of a function, the range may also be a disconnected set. The range of the function f(x) = 1 / (x² - 1), for example, is (-(, -1] ( (0, (), which does include the interval (-1, 0].

Types of Graphs

Many different kinds of graphs exist in order to present information more effectively. Common types include bar, circle, line, scatter, contour, trajectory, ternary, box and mesh plots. Each type of graph presents the function or data in a slightly different way and some graphs are more suited to a particular kind of function or data than others. Graphs can be further delineated by the type of coordinate system used to present the function or data. Use of the Cartesian coordinate system and polar coordinate systems is common.