algebra

Function

A function is a rule that assigns exactly one output to each input. Notation: f(x) = ... means "the output of f when x is the input."

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A function ff from a set AA (the domain) to a set BB (the codomain) is a rule that assigns to each element x∈Ax \in A exactly one f(x)∈Bf(x) \in B. The set of all actual outputs is the range.

Functions are the backbone of all of mathematics — calculus studies their derivatives and integrals, linear algebra studies linear functions (matrices), and computer science studies them as algorithms.

A function can be defined by:

  • A formula (f(x)=x2+1f(x) = x^2 + 1),
  • A graph (set of (x,f(x))(x, f(x)) points),
  • A table of input-output pairs,
  • Or a verbal rule.

The vertical line test distinguishes functions from arbitrary relations: any vertical line crosses the graph of a function at most once.