Domain of a function

In mathematics, the domain of definition (or simply the domain) of a function is the set of 'input' or argument values for which the function is defined. That is, the function provides an 'output' or value for each member of the domain. Conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function. In mathematics, the domain of definition (or simply the domain) of a function is the set of 'input' or argument values for which the function is defined. That is, the function provides an 'output' or value for each member of the domain. Conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function. 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. Given a function f : X → Y {displaystyle fcolon X o Y} , the set X {displaystyle X} is the domain of f {displaystyle f} ; the set Y {displaystyle Y} is the codomain of f {displaystyle f} . In the expression f ( x ) {displaystyle f(x)} , x {displaystyle x} is the argument and f ( x ) {displaystyle f(x)} is the value. One can think of an argument as a member of the domain that is chosen as an 'input' to the function, and the value as the 'output' when the function is applied to that member of the domain. The image (sometimes called the range) of f {displaystyle f} is the set of all values assumed by f {displaystyle f} for all possible x {displaystyle x} ; this is the set { f ( x ) | x ∈ X } {displaystyle left{f(x)|xin X ight}} . The image of f {displaystyle f} can be the same set as the codomain or it can be a proper subset of it; it is the whole codomain if and only if f {displaystyle f} is a surjective function, and otherwise it is smaller. A well-defined function must map every element of its domain to an element of its codomain. For example, the function f {displaystyle f} defined by has no value for f ( 0 ) {displaystyle f(0)} . Thus, the set of all real numbers, R {displaystyle mathbb {R} } , cannot be its domain. In cases like this, the function is either defined on R ∖ { 0 } {displaystyle mathbb {R} setminus {0}} or the 'gap is plugged' by explicitly defining f ( 0 ) {displaystyle f(0)} .If we extend the definition of f {displaystyle f} to the piecewise function then f is defined for all real numbers, and its domain is R {displaystyle mathbb {R} } . Any function can be restricted to a subset of its domain. The restriction of g : A → B {displaystyle gcolon A o B} to S {displaystyle S} , where S ⊆ A {displaystyle Ssubseteq A} , is written g | S : S → B {displaystyle left.g ight|_{S}colon S o B} .