Function (mathematics)

A function is a mathematical relationship between two sets of elements, where each element from the first set corresponds to exactly one element from the second set. The first set is called the domain (the input set), and the second set is called the codomain (the output set). The standard notation is f(x), where x is the input and f(x) is the result.

The essential rule of a function is that each input must produce exactly one output. If a single input produces two different outputs, that relationship is not called a function. Example: f(x) = x + 3 adds three to every number, so f(2) = 5.

Components of a function

The domain is the set of all permitted inputs. For example, the function 1/x cannot accept x = 0, because division by zero is undefined. The range is the set of actual outputs that the function produces, which may sometimes be smaller than the entire codomain.

Common types

Linear functions have the form f(x) = ax + b, and their graph is a straight line. Quadratic functions take the form ax² + bx + c, and their graph is a parabola. There are also trigonometric functions such as sine and cosine, exponential functions, and logarithmic functions. Each has its own domain and rate of change.

Important properties

A function is called injective (one-to-one) if different inputs always produce different outputs. It is called surjective if every element of the codomain can be reached. When both properties hold, the function is bijective, and it has an inverse function that maps outputs back to inputs.

Applications

Functions are the foundation of mathematical analysis, especially calculus, which studies how functions change (derivatives) and how to compute total accumulated change (integrals). In science and engineering, functions describe relationships between time, velocity, temperature, and other measurable quantities. In computer programming, the term "function" refers to a piece of code that takes input and returns output, a concept that originates from mathematics.