Arvoalueen

Arvoalueen refers to the range of values a function can output. More precisely, if f is defined on a domain D and maps to the real numbers, the arvoalueen of f is the set { f(x) | x ∈ D }. This set is also called the image of D under f. The arvoalueen is distinct from the codomain, which is the target set that contains all potential outputs by definition.

Determining the arvoalueen depends on the nature of the function and its domain. For continuous functions on an interval, the arvoalueen is typically an interval, with endpoints found from limits or the extreme value theorem. For discrete domains, the arvoalueen is a discrete set of values. In general, one analyzes how the function behaves across its domain, using monotonicity, end behavior, and solvability of equations f(x) = y to identify all possible output values.

Examples help illustrate the concept. For f(x) = x^2 with domain all real numbers, the arvoalueen is [0, ∞). For f(x) = sin x with domain all real numbers, the arvoalueen is [-1, 1]. For a linear function f(x) = 2x + 3 with domain R, the arvoalueen is (-∞, ∞).

Understanding arvoalueen aids graphing, solving equations of the form f(x) = y, and assessing invertibility. A function is invertible on a given domain if its arvoalueen matches the codomain and the function is bijective on that domain.