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

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