Domain of a Function
Definition
In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. It is the set X in the notation f: X → Y, and is alternatively denoted as dom(f). Since a (total) function is defined on its entire domain, its domain coincides with its domain of definition. However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain.
A domain is part of a function f if f is defined as a triple (X, Y, G), where X is called the domain of f, Y its codomain, and G its graph.
A domain is not part of a function f if f is defined as just a graph. For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form f: X → Y.
For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases).
If the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the x-axis.
The domain of a function f can be thought of as the set of all x values that can be plugged into f(x) that return a valid output. For example, if we have a function g in the Cartesian plane, the domain is all of the x values such that g(x) is a real number.
Examples:
- The domain of f(x) = x^2 is all real numbers, since x^2 is real for all real numbers.
- The domain of g(x) = 1/x is all real numbers EXCEPT 0, since 1/0 is not defined.
- The domain of h(x) = sqrt(x) is the set [0, inf), since sqrt(x) is only defined when x is nonnegative (that is, when x is greater than or equal to 0).
