Function

Let $X$ and $Y$ be sets. A function from $X$ to $Y$, also called a map or mapping, is a relation $G_f$ from $X$ to $Y$, also referred to as the graph of $f$, with the following two properties:

1. For every $x\in X$, there exists a $y\in Y$ such that $(x,y)\in G_f$;
2. For every $x\in X$ and all $y,z\in Y$: if $(x,y)\in G_f$ and $(x,z)\in G_f$, then $y=z$.

In other words: $f$ is a function from $X$ to $Y$ if for each element $x$ of the set $X$ there is exactly one element $y$ of $Y$ such that $(x,y)$ is an element of $G_f$. A function is often specified via a function rule; see the examples.

If $(x,y)\in G_f$, we usually denote this as $y=f(x)$ and call $y$ the image of $x$ under $f$ and $x$ an original of $y$ under $f$. We often denote a function $f$ as $f:\;X\rightarrow Y$. The set $X$ is called the domain of the function, just like with any relation, and sometimes denoted as $D_f$; $Y$ is called the codomain. The range or complete image of the function $f$ consists of the set of all images under $f$. Instead of $\mathrm{ran}(f)$, we usually write $\mathrm{im}(f)$ or $R_f$.

Let $f:\;X\rightarrow Y$ be a function from $X$ to $Y$ and $W\subset Y$. Then the preimage $f^{-1}(W)$ of $W$ under $f$ is defined as

$$f^{-1}(W)=\{x\in X\mid f(x)\in W\}\text.$$

Examples

$$\begin{aligned}&f:\;\mathbb{N}\rightarrow \mathbb{N},\\ &f(x)=\begin{cases} 0& \text{for even }x\\ 1&\text{for odd }x\end{cases}\end{aligned}$$ $$\begin{aligned}&g:\;\{a,b,c\}\rightarrow \{\alpha,\beta,\gamma\},\\ &g(a)=\alpha, g(b)=\beta, g(c)=\gamma\end{aligned}$$ $$\begin{aligned}&h:\;\{a,\alpha,b\}\rightarrow \{A,B,C\},\\ &h(a)=A, h(\alpha)=A, h(b)=B, \end{aligned}$$ $$\begin{aligned}&F:\;\mathbb{R}\backslash\{0\}\rightarrow \mathbb{R}\\ &F(x)=\frac{1}{x}\end{aligned}$$ But unlike $F$, $$\begin{aligned}&G:\;\mathbb{R}\rightarrow \mathbb{R}\\ &G(x)=\frac{1}{x}\end{aligned}$$ is not a function because the image of $0$ is not well-defined via the rule.

Equality of functions

Two functions $f$ and $g$ are equal if they are defined over the same domain and for every $x$ in the domain:$f(x)=g(x)$.
In other words, the functions $f$ and $g$ are equal if and only if their graphs are equal, i.e. $G_f=G_g$.

Sometimes it is also required that the codomain of $f$ and $g$ are equal to each other. This, for example, results in the functions $f:\;\mathbb{Z}\rightarrow \mathbb{Z}, f(x)=|x|$ and $g:\;\mathbb{Z}\rightarrow \mathbb{N}, g(x)=|x|$ not being equal.

Example

$$\begin{aligned}&f:\;\mathbb{R}\rightarrow \mathbb{R},\\ &f(x)=\sin(2x)\end{aligned}$$ and $$\begin{aligned}&g:\;\mathbb{R}\rightarrow \mathbb{R},\\ &g(x)=2\sin(x)\cos(x)\end{aligned}$$ are equal functions, specified in two ways.

Restriction of a function

Let $f:\;X\rightarrow Y$ be a function from $X$ to $y$ and $W\subset X$. The restriction of $f$ to $W$ is the function $f{\restriction}_{W}:\; W\rightarrow Y$ defined by $f{\restriction}_W(x)=f(x)$ for all $x\in W$. In other words, as a relation: $G_{f{\restriction}_W}=\bigl\{(x,y)\in G_f\mid x\in W\bigr\}$.

The image of $W$ under $f$ is $\{f(x)\mid x\in W\bigr\}$.

Examples

$$\begin{aligned}&f:\;\mathbb{Z}\rightarrow \mathbb{Z},\quad f(n)=\begin{cases} 0 &\text{if }n<0\\ 1&\text{if }n\ge 0\end{cases}\\[0.25cm]&f{\restriction}_\mathbb{N}:\;\mathbb{N}\rightarrow \mathbb{Z}, \quad f{\restriction}_\mathbb{N}(n)=1\end{aligned}$$
$$\begin{aligned}&g:\;\mathbb{R}\rightarrow \mathbb{R},\quad g(x)=\cos \pi x\\[0.25cm]&g{\restriction}_\mathbb{Z}:\;\mathbb{Z}\rightarrow \mathbb{R}, \quad g{\restriction}_\mathbb{Z}(n)=(-1)^n\end{aligned}$$