A function in math is a rule (made up of a set of operations on a number) that describes how a given number may be transformed.
The expression can be described in the format:
Which can be written more explicitly:
A rule may be described by a mapping of inputs to its outputs:
The set of valid input numbers of a function is called its domain while the set of possible output numbers is called its codomain.
Domain and codomain of a function are not always explicit.
Functions may be written in a number of ways:
Ambiguous notation that implicitly uses the set of real number for its domain and codomain.
Given the function definition
becomes shorthand for
is the image of x of
This notation is ambiguous because could refer to or an image of for a certain number if one has previously been declard for .
Also implies a domain and codomain of . This notation is used to define rules inline without naming them.
Is often used instead of functional notation. It uses a subscript instead of parantheses to name its input.
This notation is typically used for sequences of natural numbers.
The output of one function can be directly channeled into the input of another function by combining their letters.
(the innermost function is applied first)
An inverse function runs a function's opposite operations in reverse order.
The inverse to is
Finding the input of a given output
If a function's rule and an output are known, that output can be traced through the function's inverse rule to find its original input (domain).