Definition 5.27 (Domain, Codomain, Image, Preimage)

#Definition

Let $f : A \to B$ be a function

Then the set $A$ is called the $domain$ of the function $f$ , and the set $B$ is called the $codomain$ of the function $f$

If $f (a) = b$ , we say that $b$ is the $image$ of $a$ , and $a$ is the $preimage$ of $b$

The $range$ (also called $image$ ) of $f$ is the set of all images of elements of $A$

Also if $f$ is a function from $A$ to $B$ , we say the $f$ $maps$ $A$ to $B$

Example

What is the domain, codomain and range of the function that assigns grades to the students of our module?

$\color{lightgreen}\text{The domain is thesetof students in our module}$
$\color{lightgreen}\text{The codomain is thesetof all possible grades}$

$\color{lightgreen}\text{The range is thesetof those grades,}$
$that have been assigned to at least one student$

When specifying a function, you $always$ need to say

Which set you choose as the $domain$ ,
Which set you choose as the $codomain$ , and
$For each memeber of the domain$ you have to say $which member of the codomain it is mapped to$

Important

Note that if you change the domain or the codomain, you have a different function.
If you change the mapping of the elements, you also have a different function.