What is a function?

In elementary algebra, functions were presented as formulas such as $f(x) = 8x^3 + 31$. In computer science, functions take an input, performs a calculation with the input, and outputs the result. These two ideas are related; functions can be represented as formulas and they associate an input $x$ to a single output $f(x)$. There are three main parts to a function:

1. The input
2. The relationship
3. The output

For a squaring function, the relationship is such that the output is the square of the input. It can be represented as $f(x) = x^2$, where $f$ is the function name, $x$ is the input, and $x^2$ is the output. It is simply convention to use $f$ and $x$; we can use different function names (or even no name at all) and different symbols. The general idea is that a function relates an input to an output. For example, a tree grows 20 cm every year, so its height is related to its age; we can define this with the height function:

$$height(age) = age × 20$$

If the tree is 10 years old, then the height is:

$$height(10) = 10 × 20 = 200 cm$$

However, there is a limit to what can be used as an input to a function; a set is used to specify what a function takes as input (and a set is used to specify what a function can output). The domain is all the values that can go into a function and the range or image is all the values that are outputted. An additional term is the codomain, which are the values that may be possibly outputted. The notation is:

$$f : X \rightarrow Y$$

where $X$ is the domain, $Y$ is the codomain, and $f(X)$ is the range.

Furthermore, functions have special rules:

1. It must work for every possible input value
2. It has only one relationship for each input value

A function relates each element of a set with exactly one element of another set; a one-to-many relationship is not allowed, however, a many-to-one relationship is allowed.

Further reading at http://en.wikibooks.org/wiki/Linear_Algebra/Sets,_Functions,_Relations