Definitionsmenge

Definitionsmenge is the German term for the domain of definition of a function. It denotes the set of all input values for which the function is defined and yields an output that lies in its codomain. The definitionsmenge is a subset of the input space and is essential for understanding where a function or expression can be evaluated.

Formally, if a function f is defined on a set X and maps to a codomain Y, the definitionsmenge D can be described as D = { x in X | f(x) is defined and f(x) ∈ Y }. In many contexts, the definitionsmenge is referred to as the domain (Definitionsbereich) of the function.

The definitionsmenge is determined by the mathematical form of the expression. Constraints such as denominators that must not be zero, radicands that must be nonnegative (for real-valued functions), and arguments of logarithms that must be positive restrict the domain. Piecewise definitions can also yield a definitionsmenge consisting of the union of the domains of the individual pieces.

Examples:

- f(x) = 1/x has definitionsmenge all real numbers except x = 0.

- f(x) = sqrt(x) has definitionsmenge [0, ∞) in the real numbers.

- f(x) = sqrt(1 − x^2) has definitionsmenge [−1, 1].

The definitionsmenge is related to, but distinct from, the range (the set of actual outputs) and the codomain (the target set). It is also related to the concept of the domain of a function in more general settings, including complex numbers and abstract spaces. See also Domain, Range, Codomain, Preimage.