Function

A function is a many(one)-to-one map between two sets.

Formal definition

Let $D_f,C_f,R_f \neq \emptyset$ be sets.

If

$$ \forall x \in D_f: \exists y \in C_f, f(x)=y, $$

then $f$ is a function of domain $D_f$ and codomain $C_f$, noted as $f: D_f \to C_f$;

Furthermore,

if

$$ \forall y \in R_f \subseteq C_f, \exists x \in D_f, f(x)=y, $$

then $R_f$ is the range of $f$;

Lastly,

if

$$ \forall y \in R_f: \exists f^{-1}, f^{-1}(y)=x, $$

then function $f$ is said to be reversible and the bijection of $D$ to $R_f$. $f$ and $f^{-1}$ are said to be inverse functions of each other.

While a function can be many-to-one, a bijection is an one-to-one function of its domain to its range.

Example:

$$ f(x)=2x^2, x \in \mathbb{R} $$

is a function, where $f: \mathbb{R} \to \mathbb{R}$.

It is however not a bijection from $\mathbb{R}$ to $\mathbb{R}$, as $\exists y \in \mathbb{R}, \nexists f^{-1}, f^{-1}(y)=x$, such as $y=-1$

Note the range $R_f \subset \mathbb{R}$, as $\exists y \in \mathbb{R}: y \notin R_f$, such as $y=-1$

Image and preimage

Given function $f: A \to B$

If for $A_1 \subseteq A, B_1 \subseteq B$,

$$ B_1=f(A_1)=\{f(a)|\forall a \in A_1\} $$

Then $B_1$ is known as the image of $A_1$ in $B$ under $f$, and $A_1$ is known as the pre-image of $B_1$ under $f$, noted $f^{\leftarrow}(B_1)=A_1$