function

a function (or mapping) $\text{[math]}$ from a set $\text{[math]}$ to a set $\text{[math]}$ is a rule that assigns to each element $\text{[math]}$ of $\text{[math]}$ exactly one element $\text{[math]}$ of $\text{[math]}$ . the set $\text{[math]}$ is called the domain of $\text{[math]}$ , and $\text{[math]}$ is called the range of $\text{[math]}$ . if $\text{[math]}$ assigns $\text{[math]}$ to $\text{[math]}$ , then $\text{[math]}$ is called the image of $\text{[math]}$ under $\text{[math]}$ . the subset of $\text{[math]}$ comprising all the images of elements of $\text{[math]}$ is called the image of $\text{[math]}$ under $\text{[math]}$ .
[cite:;from @abstract_gallian_2021 chapter 0 preliminaries; definition function (mapping)]

a function that is assumed total unless stated otherwise (partial).

alternative definitions of functions

let $\text{[math]}$ be sets, and let $\text{[math]}$ be a property pertaining to an object $\text{[math]}$ and an object $\text{[math]}$ , such that for every $\text{[math]}$ , there is exactly one $\text{[math]}$ for which $\text{[math]}$ is true (this is sometimes known as the vertical line test). then we define the function $\text{[math]}$ defined by $\text{[math]}$ on the domain $\text{[math]}$ and codomain to be the object which, given any input $\text{[math]}$ , assigns an output $\text{[math]}$ , defined to be the unique object $\text{[math]}$ for which $\text{[math]}$ is true. thus, for any $\text{[math]}$ and $\text{[math]}$ ,
$\text{[math]}$ [cite:;taken from @tao_analysis_1 definition 3.3.1]

a function $\text{[math]}$ is a rule of assignment $\text{[math]}$ , together with a set $\text{[math]}$ that contains the image set of $\text{[math]}$ . the domain $\text{[math]}$ of the rule $\text{[math]}$ is also called the domain of the function $\text{[math]}$ ; the image set of $\text{[math]}$ is also called the image set of $\text{[math]}$ ; and the set $\text{[math]}$ is called the range of $\text{[math]}$ .
[cite:;taken from @topology_munkres_2014 chapter 1 set theory and logic]