Function (mathematics)

In mathematics, a function is a mathematical object that produces an output, when given an input (which could be a number, a vector, or anything that can exist inside a set of things).

In essence, a function is like a machine, that takes a value of ${\displaystyle x}$ and returns an output ${\displaystyle y}$. The set of all values that ${\displaystyle x}$ can have is called the domain, and the set that contains every value that ${\displaystyle y}$ can have is called the codomain. A function is often denoted by italic letters such as ${\displaystyle f}$, ${\displaystyle g}$, ${\displaystyle h}$.^[1][2][3]

If this happens, then we say that ${\displaystyle y}$ is a function of ${\displaystyle x}$, and we write ${\displaystyle y=f(x)}$. Here, ${\displaystyle f}$ is the name of the function, and one writes ${\displaystyle f:X\to Y}$ (function from X to Y) to represent the three parts of the function: the domain (${\displaystyle X}$), the codomain (${\displaystyle Y}$), and the pairing process (the arrow), or, the mapping from and to the sets.

An example of a function is ${\displaystyle f(x)=x+1}$. One gives a natural number ${\displaystyle x}$ as the input, and gets a natural number ${\displaystyle y}$, which is ${\displaystyle x+1}$. For example, giving 3 as input to ${\displaystyle f}$ results in an output of 4.

A function doesn't have to be an equation. The main idea is that inputs and outputs are paired up somehow—even if the process might be very complicated.

1. ↑ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-17.
2. ↑ "What is a Function". www.mathisfun.com. Retrieved 2020-08-17.
3. ↑ Weisstein, Eric W. "Function". mathworld.wolfram.com. Retrieved 2020-08-17.