User:Frumfst/Functionality

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This page is for a thought up math term which will automatically be deleted if put on an actual Wikipedia page.

Functionality is the coined term for an order of a function which describes the extent (and direction) the original function is performed. For example, take the function f(x)=x+4. The second (functionality) order of the function would be f(f(x))=x+8; the original function is performed twice. The third order of the function would be f(f(f(x)))=x+12; the original function is performed three times.

The functionality order of a function does not have to be positive. For example, the inverse function of f(x) would simply have the order of negative one (-1); the function is performed once in the negative (or undoing) direction. An order of negative two (-2) would indicate that the inverse function is performed twice.

Neither does the order have to be an integer. For example, take the function f(x)=x64. The one-half (.5) order functionality g(x) would equal x8 because g(g(x))=x64; g(x)=x8 must be compounded twice to equal the original function. The one-third (1/3) order functionality g(x) would equal x4 because g(g(g(x)))=x64. Such functionalities where the order of the function is equal to one divided by a positive integer (1/n) are called "functionality factors" of the original equation. This is because you can compound them an integer number of times to get the original function.

The functionality is denoted by a subscript number by the function. For example, the .5 functionality order of h(x) would be denoted as f.5(h(x)). This would mean that the answer function g(x) would be such that g(g(x)) is equal to the original function h(x).

As of July 14, 2007, the notation is no longer confusing, the correct way is f2(x2)=x4; the equation f2(x4)=x2 is no longer correct.

The first 4 functionality factors of the function f(x)=3x+2 and the limit of the function as the order approaches zero. In this case the zeroeth order is the line y=x, the line at which the centripit is located.
The first 4 functionality factors of the function f(x)=3x+2 and the limit of the function as the order approaches zero. In this case the zeroeth order is the line y=x, the line at which the centripit is located.

[edit] Functionality rules

fo(x) is the oth order functionality of x.

f_o(f_p(g(x)))=f_{o+p}(g(x))\,

f_0(g(x))=x\,

f_o(f_{-o}(g(x)))=f_{-o}(f_o(g(x)))=x\,

Simple Addition Rule: f_o(x+B)=x+B o\,

Simple Multiplication Rule: f_o(Ax)=A^{o}x\,

Simple Power Rule: f_o(x^n)=x^{n^{o}}\,

Addition Rule for a power of 1: f_o(Ax+B)=A^{o}x + B({\frac{a^{o}-1}{a-1}})

Multiplication Rule for a power of n: f_o(Ax^n)=A^{\frac{n^{o}-1}{n-1}} x^{n^{o}}

Polynomial Approximation Rule: f_o(p(x)) \approx x^{n^{o}} ({\frac{p(x)}{x^{n}}})^{\frac{n^{o}-1}{n-1}}

Error of Polynomial Rule: Error≤B (with x close to 0) with greatest error at x=0

[edit] The Centripit

All the functionality "factors" (any order where o equals one divided by an integer) of one function intersect at a point called the centripit. The centripit lies on the line y=x and can quite easily be found for any linear function. There is one exception: any function parallel to the function f(x)=x. Every functionality order of this line is parallel (or equal) to y=x. In this exception, there is no centripit as it intercepts the line y=x at all points or none at all. However, any function parallel to the function f(x)=x will approach the line y=x as the order of the functionality approaches zero.