function composition

consider 2 functions $\text{[math]}$ , the function from $\text{[math]}$ to $\text{[math]}$ that is denoted as $\text{[math]}$ where $\text{[math]}$ for all $\text{[math]}$ , is called the composite function of $\text{[math]}$ over $\text{[math]}$ .

an equivalent definition from gallian's:

let $\text{[math]}$ and $\text{[math]}$ . the composition $\text{[math]}$ is the mapping from $\text{[math]}$ to $\text{[math]}$ defined by $\text{[math]}$ for all $\text{[math]}$ in $\text{[math]}$ .
[cite:;from @abstract_gallian_2021 chapter 0 preliminaries; definition composition of functions]

$\text{[math]}$

$\text{[math]}$ and $\text{[math]}$ arent necessarily equal

associativity
if $\text{[math]}$ then this equality holds true:
$\text{[math]}$ $\text{[math]}$
$\text{[math]}$

$\text{[math]}$ are functions therefore $\text{[math]}$ is a function
$\text{[math]}$ are functions therefore $\text{[math]}$ is a function
therefore $\text{[math]}$ and $\text{[math]}$ are functions and their domain and range are equal
let $\text{[math]}$ , then:
$\text{[math]}$

let $\text{[math]}$ be functions:

if $\text{[math]}$ are surjective, then $\text{[math]}$ is surjective
if $\text{[math]}$ are injective, then $\text{[math]}$ is injective
if $\text{[math]}$ are total, then $\text{[math]}$ is total

$\text{[math]}$ being surjective or injective doesnt necessarily mean $\text{[math]}$ or $\text{[math]}$ are too