Definitions

A function from ${\displaystyle A}$ to ${\displaystyle B}$ is set of ordered pairs ${\displaystyle S\subseteq A\times B}$ such that: For sets ${\displaystyle A}$ and ${\displaystyle B}$, "${\displaystyle f:A\to B}$" means ${\displaystyle f}$ is a function from ${\displaystyle A}$ to ${\displaystyle B}$. If ${\displaystyle f}$ is a function, then "${\displaystyle f\left(u\right)=v}$" means ${\displaystyle (u,v)\in f}$.

The set ${\displaystyle A}$ is called the function's domain and the set ${\displaystyle B}$ is called the function's codomain. The above definition basically says that a) for every element ${\displaystyle u}$ in ${\displaystyle A}$, ${\displaystyle f\left(u\right)\in B}$, and b) if ${\displaystyle f\left(u\right)=v\text{and}f\left(u\right)=w}$, then ${\displaystyle v=w}$ (ie, each element of the function's domain maps to exactly one element of the function's codomain). The word "range" is sometimes used to refer to a function's codomain, and sometimes to refer to its image set.

The image set of a function is the subset of the function's codomain onto which elements of its domain are mapped.

The image set of a function ${\displaystyle f:A\to B}$, is defined as and may be written as ${\displaystyle Im\left(f\right)\text{or}f\left(A\right)}$.

A function is surjective if the function's image set equals its codomain.

A function ${\displaystyle f:A\to B}$ is surjective (or "onto") if More concisely, ${\displaystyle f}$ is surjective if ${\displaystyle Im\left(f\right)=B}$. This essentially means that every element of the function's codomain is mapped to by some element of its domain.

A function is injective (or "one-to-one") if no element of its codomain is mapped to by more than one element of its domain.

A function ${\displaystyle f:A\to B}$ is injective if This means that no element of the function's codomain is mapped to by more than one element of its domain. In order for a function to be invertible, it must be one-to-one.

A function is bijective if it is both surjective and injective.

Given ${\displaystyle f:A\to B}$ and ${\displaystyle g:B\to C}$, the composition of ${\displaystyle f}$ and ${\displaystyle g}$ (written ${\displaystyle g\circ f}$ and read "${\displaystyle g}$ composition ${\displaystyle f}$" or "${\displaystyle g}$ of ${\displaystyle f}$") is the subset of ${\displaystyle A\times C}$ equal to In other words, ${\displaystyle (u,w)\in g\circ f}$ means ${\displaystyle w=g\left(f\left(u\right)\right)}$, and ${\displaystyle \left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)}$ for all ${\displaystyle x}$.

For a function ${\displaystyle f}$ with domain ${\displaystyle ℝ}$, ${\displaystyle f}$ is strictly increasing if: