function

A function $f(x)$ is a set $S$ of ordered pairs that map the first value of the ordered pair to the second value in the ordered pair, where the first value may not have duplicates in the set $S$. The map from the first value to the second value has the notation $f(x) = y$ for some x and y, where $f$ is the mapping. Note that we can also define rules for $f$ and do not therefore have to explicitly define all the mappings:

Which is an example of a parabolic function. $x$ and $y$ can both conceptually be any object, but usually they are mathematical objects. Some examples of such objects include tensors and scalars.

However, we must find a way to define what an ordered pair is. Sets have no order by default, so we need to add order by doing the following:

Where the element that is not explicitly a set gives us the definition of the first element.

Let $(S, \circ)$ define a group where $S$ is the set of all functions, and $\circ$ is the composition binary operator. Then $f(x) = x$ is the identity element, and an inverse of a function is defined as $(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x$.