Function (mathematics)

Function machine2

Example Function

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Binary operations as black box

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Motor vehicle deaths in the US

Motor vehicle deaths in the US histogram

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Function (mathematics)

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example of a function is the relation that assigns to each real number x the real number x^2. The term "function" was first introduced by Gottfried Wilhelm Leibniz, a German mathematician and philosopher, in the 17th century. Functions are fundamental to the study of calculus, algebra, and mathematical analysis, and they are used to describe mathematical relationships and changes between quantities.

Definition

A function f from a set X (the domain) to a set Y (the codomain) is defined by a set of ordered pairs (x, y) where x is in X, y is in Y, and no two different pairs have the same first element. This can be symbolically represented as f: X → Y. The set X is known as the domain of the function, the set Y is known as the codomain, and the set of all y such that there exists an x in X with (x, y) in the function is called the range.

Types of Functions

Functions can be classified in various ways, based on their properties:

• Injective (One-to-One): A function is injective if different inputs produce different outputs.
• Surjective (Onto): A function is surjective if for every element in the codomain, there is at least one element in the domain that maps to it.
• Bijective (One-to-One Correspondence): A function is bijective if it is both injective and surjective. This means there is a perfect pairing between the domain and codomain.
• Continuous: In calculus, a function is continuous if, intuitively, small changes in the input result in small changes in the output.
• Differentiable: A function is differentiable if it has a derivative at each point in its domain.

Examples

• The function f(x) = x^2 maps each real number to its square.
• The function f(x) = 2x + 1 is a linear function, mapping each real number to a value twice itself plus one.

Importance in Mathematics

Functions are a central concept in mathematics due to their ability to model relationships between quantities and their changes. They are used in virtually every branch of mathematics and are the foundational concept in calculus and mathematical analysis. Functions allow mathematicians and scientists to describe the world in terms of mathematical equations, making it possible to predict, understand, and control various phenomena.

See Also

• Graph of a function
• Function composition
• Inverse function
• Limit (mathematics)
• Derivative