About

A function $f$ defined on a set $X$ is an assignment of an element in some set $Y$ to each element of $X$. The set $X$ is called the domain of the function and $Y$ is called the codomain. The elements of $X$ are the inputs to the function and the elements of $Y$ are the potential outputs. For some input $x \in X$, its corresponding output in $Y$ is denoted $f(x)$. Not every element of $Y$ needs to be the output corresponding to some input though: the subset of $Y$ containing the elements that are an output of the function is called the range of $f$. When a function $f$ has domain $X$ and codomain $Y$, this is signified by writing $f \colon X \to Y$, and the assignments of inputs to outputs is signified by writing $f\colon x \mapsto f(x)$.

If you have a function whose codomain is the domain of another function, you can compose those two functions. In symbols if you have a function $f\colon X \to Y$ and a function $g \colon Y \to Z$, their composite is a function $g\circ f\colon X\to Z$ defined by the assignment $g\circ f\colon x \mapsto g(f(x))$.

For many examples of functions, the domain and range of the function are topological spaces, meaning that they are equipped with some notion of geometry. In this case we like to think of the function $f\colon X\to Y$ geometrically as the subset of the points $(x,f(x))$ in the topological space $X \times Y$. This subset of all the input-output pairs is called the graph of $f$.

Often mathematics textbooks will define a function slightly more rigorously than this though. They'll say that a function $f \colon X \to Y$ is a relation $R$ on the set $X \times Y$ such that

1. For each $x \in X$ there is some $y \in Y$ such that $xRy$. Each input needs an output.
2. If $xRy$ and $xRz$, then $y=z$. Each input needs exactly one output.

Here are a bunch of examples of functions:

• Many examples of functions covered in elementary and high school have as their domain and codomain the real numbers $\mathbf{R}$. A basic example is the function $f \colon \mathbf{R} \to \mathbf{R}$ defined by the rule $f(x) = x^2$. Thinking geometrically, the graph of $f$ is the set of all points $(x,x^2)$ in the plane $\mathbf{R}^2$, and this forms a parabola. Note that while the codomain of this function is $\mathbf{R}$, the range consists of only the non-negative real numbers.

• Here's a silly example. For any set $X$ we can define an identity function $\mathbf{1}_X$ with domain and codomain $X$ such that $\mathbf{1}_X \colon x \mapsto x$.

• Let $W$ denote the set of all strings of letters of the alphabet, so like $\text{npr}$ or $\text{asdfasdf}$ or $\text{butt}$ for example. And let $\mathbf{N}$ denote the set of natural numbers. We can define a function $\ell\colon W \to \mathbf{N}$ such that $\ell$ assigns to each word it's length. So $\ell(\text{defenestration}) = 14$. Also $\ell(\text{butt})=4$.

• Using the same set $W$ in the last example, let's define another function $\tau\colon W \to W$ such that $\tau$ "reverses" a word. So $\tau(\text{defenestration}) = \text{noitartsenefed}$, and $\tau(\text{butt}) = \text{ttub}$. A few neat properties of $\tau$ that deserve to be pointed out, $\tau \circ \tau = \mathbf{1}_W$, and also $\ell\circ\tau = \ell$.