Arvojoukkoa

Arvojoukkoa is a Finnish mathematical term describing the range, or image, of a function. When a function f maps a domain X to a codomain Y, the arvojoukko of f consists of all values y in Y that f attains for some x in X.

Formally, for a function f: X → Y, arvojoukko(f) = { f(x) | x ∈ X }. In many contexts it is called the image of f, and it represents the actual outputs produced by the function, as opposed to the possible outputs defined by the codomain.

Examples help illustrate the concept. If f: R → R is defined by f(x) = x^2, the arvojoukko is [0, ∞). If f: Z → Z is defined by f(n) = n mod 3, the arvojoukko is {0, 1, 2}. If f: R → R is defined by f(x) = e^x, the arvojoukko is (0, ∞).

Properties of arvojoukko reflect the mapping’s reach. The arvojoukko equals the codomain Y exactly when f is surjective onto Y. If f is not surjective, arvojoukko is a proper subset of Y. If f is constant, arvojoukko is a singleton set containing that constant value.

In Finnish mathematics education, arvojoukko is a standard term for teaching the concept of the range of a function. It is closely related to other notions such as domain (the input set) and image, which is another common term for the arvojoukko in English-language texts.