inverse function

functions $\text{[math]}$ which are both one-to-one and onto are also called bijective or invertible. if $\text{[math]}$ is bijective, then for every $\text{[math]}$ , there is exactly one $\text{[math]}$ such that $\text{[math]}$ (there is at least one because of surjectivity, and at most one because of injectivity). this value of $\text{[math]}$ is denoted $\text{[math]}$ ; thus $\text{[math]}$ is a function from $\text{[math]}$ to $\text{[math]}$ . we call $\text{[math]}$ the inverse of $\text{[math]}$ .
[cite:@tao_analysis_1 definition 3.3.23]

note that usually we propose the definition of an invertible function, propose the definition of a bijective function, then prove that a function is invertible iff it is bijective (inverse_function.html), so while bijective functions and invertible functions imply the same thing, they have different definitions.
tao's definition seems to be considering them one and the same, which i think may be a less popular approach.

another definition from college:

a function $\text{[math]}$ is invertible if the inverse relation $\text{[math]}$ is a function
meaning if for every $\text{[math]}$ there exists a single $\text{[math]}$ such that $\text{[math]}$
meaning if for every $\text{[math]}$ there exists a single $\text{[math]}$ such that $\text{[math]}$
the function $\text{[math]}$ is called the inverse function of $\text{[math]}$

because $\text{[math]}$ , if $\text{[math]}$ is invertible, then $\text{[math]}$

let $\text{[math]}$ be a function
if $\text{[math]}$ is invertible, then $\text{[math]}$ is invertible and $\text{[math]}$

let $\text{[math]}$ be a function
$\text{[math]}$ is invertible if and only if $\text{[math]}$ is total, surjective and injective.

if $\text{[math]}$ are invertible functions then $\text{[math]}$ is invertible and $\text{[math]}$

if $\text{[math]}$ is invertible then
$\text{[math]}$ where $\text{[math]}$ is the identity function of $\text{[math]}$ and $\text{[math]}$ is the identity function of $\text{[math]}$

let $\text{[math]}$ then $\text{[math]}$

the composition of invertible functions results in an invertible function

let $\text{[math]}$ be total functions
if $\text{[math]}$ is injective then $\text{[math]}$ is injective
if $\text{[math]}$ is surjective then $\text{[math]}$ is surjective
if $\text{[math]}$ is injective and $\text{[math]}$ is surjective then $\text{[math]}$ are all invertible (what a life-saver that theorem is!)

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

if $\text{[math]}$ is injective and $\text{[math]}$ is surjective then $\text{[math]}$ is injective
if $\text{[math]}$ is surjective and $\text{[math]}$ is injective then $\text{[math]}$ is surjective