f(x) — function notation, meaning and common uses

Formal meaning and notation

Formally a function is a mapping from a set called the domain to another set called the codomain. A compact way to record this is f: X → Y with the definition f(x) = expression. The domain is the set of x for which f(x) is defined; the range (or image) is the set of values actually attained. Notation varies: g(t), h(z) or simply f can be used instead of f(x) when context makes the argument clear.

Common examples

• Polynomial: f(x) = x^2 + 2x + 1
• Trigonometric: f(x) = sin(x)
• Exponential and logarithmic: f(x) = e^x, f(x) = log(x) (with restricted domain)
• Piecewise definitions and functions on discrete or abstract sets

Operations and properties

Functions can be composed (f∘g)(x) = f(g(x)), inverted when one-to-one, added, multiplied, differentiated and integrated when appropriate. Important properties include continuity, monotonicity, periodicity and differentiability; these describe behavior rather than change the basic meaning of f(x).

History and other uses

The idea of a function evolved through 17th–18th century mathematics; the compact notation f(x) became standard in later centuries and is often attributed to the influence of Euler and other classical analysts. Outside mathematics, the form f(x) appears informally in programming and in popular culture as a label; for example, it is sometimes used as a stylized name in music and branding.