A function is a mathematical expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable) . It is a relationship between inputs where each input is related to exactly one output. A function has three parts: a set of inputs, a set of outputs, and a rule that relates the elements of the set of inputs to the elements of the set of outputs in such a way that each input is assigned exactly one output. The modern definition of a function was first given in 1837 by the German mathematician Peter Dirichlet.
Key features of a function include:
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Domain: The set of values of the independent variable for which the function is defined.
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Range: The set of values of the dependent variable that the function can take.
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Notation: Functions are commonly denoted by a one-letter name, such as f, g, or h, and are often written as y = f(x) or f: X → Y, where X is the domain and Y is the codomain or range.
Functions can be defined by a formula that describes a combination of arithmetic operations and previously defined functions. A function machine is a useful metaphor for understanding functions, where an input is fed into the machine and an output is produced based on the input.
