Function

A function is a process that converts units of one mathematical notion into units of another.

The notation $f:A\rightarrow B$ is shorthand for the following statement: $A$ and $B$ are some mathematical notions, $f$ is a function converting an $A$ into a $B$.

Examples

Figure (1): Redstone XOR gate in Minecraft

Functions

The following table encodes a function:

f(x)

x	(0,0)	(1,0)	(0,1)	(1,1)
f(x)	0	1	1	0

That table describes the input and output of an XOR gate. A XOR gate (another example of something that performs a function) can be seen in figure (1). The following lines of C++ code perform a function:

Figure (2): A 1619 book of mathematical tables by Matthias Bernegger, showing some of the sine, tangent, and secant functions.

string reverseString(const string &s) { 
  string returnStr = ""; 
  for (int i = s.size() - 1; i >= 0; --i) { 
    returnStr.append(s[i]);
  }
  return returnStr;
}

A sequence is a function.

A table describing several trigonometric functions is shown in figure (2).

A graph can describe a function: it provides instructions for producing outputs from inputs.

The method that an elementary schooler learns for carrying out long division is a function.

Non-functions

A coffee machine is not a function. It carries out a process; it takes some inputs (energy, water, coffee grounds, paper filter) and converts them into coffee. But the inputs and outputs are not mathematical notions.

The traditional concept

In standard mathematics, a function always converts elements of a set $X$ into elements of a set $Y$. The sets could be infinite, so the function doesn't have to describe a real process. In fact, though standard mathematicians usually think about functions as if they were processes, standard mathematics technically doesn't define functions as processes at all. Rather, it defines a function from $X$ to $Y$ as a subset of $X\times Y$ satisfying certain conditions.

A function doesn't have to describe a real process, beca