A function is a relation between a set of inputs and a set of possible outputs, where each input maps to exactly one output.
A function, denoted f(x), takes an input x and produces a unique output. Functions can be represented as equations, graphs, tables, or mappings. Key types include linear, quadratic, polynomial, exponential, logarithmic, and trigonometric functions. Functions model cause-and-effect relationships in science, economics, and engineering. The domain is the set of all possible inputs, and the range is the set of all possible outputs.
The concept of a function evolved over centuries. Gottfried Leibniz first used the word "function" in 1673. Leonhard Euler introduced the notation f(x) in 1734. The modern formal definition as a set of ordered pairs was developed by Dirichlet and Riemann in the 19th century. Functions became central to mathematics after the development of calculus.