Range (mathematics) - Citizendia

For other uses, see range.

In mathematics, the range of a function is the set of all "output" values produced by that function. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Output is the term denoting either an exit or changes which exit a System and which activate/modify a Process. Sometimes it is called the image, or more precisely, the image of the domain of the function. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined Range is also occasionally used to indicate the difference between the largest and smallest numbers in a set of real-valued data. In Mathematics, the real numbers may be described informally in several different ways If f is a surjection then its range is equal to its codomain. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, the codomain, or target, of a function f: X → Y is the set In a representation of a function in a xy Cartesian coordinate system, the range is represented on the ordinate (on the y axis). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

Examples

Let f be a function on the real numbers $f\colon \mathbb{R}\rightarrow\mathbb{R}$ defined by $f (x) = 2 x$ . In Mathematics, the real numbers may be described informally in several different ways This function takes as input any real number and multiplies it by two. Multiplication by any real number can yield any real number, therefore f(x) may be any real number, which is to say that the range of f is (-∞, ∞).

In other instances, range may be restricted by the domain of the function. Consider the function g such that $g\colon \mathbb{R}^+\rightarrow\mathbb{R}, g(x) = 2x$ . Since the codomain is $\mathbb{R}$ , any real number is a legal value for g(x). In Mathematics, the codomain, or target, of a function f: X → Y is the set However, the domain of g is $\mathbb{R}^+$ , so the input for g may only be a real number greater than zero. Multiplying any positive real number by two will always yield another positive real number, so the range of g is [0, ∞). Note here that the range of the function is not equal to its codomain, though the range is (and always will be) a subset of the codomain.

Range may also be restricted by the definition of the function. Consider the function h such that $h\colon \mathbb{R}\rightarrow\mathbb{R}, h(x) = x^2$ . Here the input is again any real number, though squaring any real number will never yield a negative number, and so the output of h may be any nonnegative number (including zero), thus the range is [0, ∞).

Examples

See also