clean up & uniformity
A '''function''' "''f''" is a fixed method for calculating a unique output "''f(x)''" for every given input "''x''". For example, the [[polynomial]] function <math>f(x) = x^2</math> takes numbers as inputs
, and outputs their squares. So in this case, if 1 is input, 1 is output, whilst if 2 is input, 4 is output.
A function is typically written with a formula for the output, in terms of the input. For example, in the case of f above, the formula is <math>f(x) = x^2</math>. When we want to input a particular value, we insert it in the place of x. So <math>f(3) = 3^2 = 3*3 = 9</math>.
i(x) = x is another function that does nothing to x. So i(1) = 1, i(2) = 2, and so on. This is called the '''identity function'''.
Some functions do not have numbers as outputs. For example, we can define a function "pres" that takes a number n, and returns the name of the nth [[
President_of_the_United_States_of_America| American President]]. So, pres(1) = [[George Washington]] (the first president), pres(2) = [[John Adams]] (the second president), and so on.
Similarly, not all functions take numbers as inputs. For example, we could define a function "presnumber" that takes the name of an American President (the nth in sequence), and outputs n. Thus presnumber(George Washington) = 1, presnumber(John Adams) = 2. This would be the inverse of pres defined above. (However, due to a historical curiosity, this function is not properly defined; see below.)
There are functions that take objects other than numbers as inputs. For example, rotation can be thought of as a function that takes a shape as an input, and outputs that shape rotated by a certain angle.
*[http://mathworld.wolfram.com/Function.html Function -- from Wolfram MathWorld]