Function

A function assigns exactly one output to each input in its domain. Its domain specifies the allowed inputs, while its codomain specifies the possible outputs.

Domain: The set of inputs for which a function is defined. A function assigns an output to every input in its domain.
Function notation: A notation such as f(x) that names a function's output for a specified input. It writes an input-output assignment compactly.
Partial function: A rule that assigns at most one output to each input but may leave some inputs unassigned. Unlike a function on a stated domain, it need not assign outputs throughout that domain.
Linear function: A function of the form f(x)=mx+b over the real numbers. Its constant rate of change makes it a basic model of proportional and affine relationships.
Functional programming: A programming paradigm that builds computation by evaluating and composing functions. It treats functions as central units of program structure and transformation.
Codomain: The specified set in which a function's outputs are required to lie. It sets the allowed output set, even when some elements are never produced.
Graph of a function: The set of coordinate points formed by pairing each input with its output. A graph displays the assignments and can be tested with the vertical line test.
Multivalued function: A relation that can assign multiple outputs to a single input. It violates the one-output requirement that defines ordinary functions.
Exponential function: A function in which the variable appears in the exponent, commonly written f(x)=a^x for a>0. It models growth or decay at a rate proportional to the current amount.
Lambda calculus: A formal system for expressing computation through function abstraction and application. It gives functions a precise role in the foundations of computation.

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