Home

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

Consider the spaces X and Y whose elements are represented by the variables x and y respectively. *A function is a mathematical relationship between X and Y that either maps X
onto Y, or maps X into Y, such that the matching of the elements of X to the elements of Y form a unique pair (x,y), called an ordered pair *.
A mapping of X *onto* Y means that every element of X has been matched with every element of Y. A mapping of X *into* Y
means that only a subset of Y has been matched with the elements of X. The set of the elements that is mapped *onto* or *into * Y is called
the **domain** of the function, and the set of elements in Y that is matched with the domain of the function is called
the **range** of the function. Y is sometimes called the **codomain** of the function
and the ordered pair (x,y) is sometimes called the **graph** of the function. The variable *x* is called the **independent variable**
while the variable *y* is called the **dependent variable** because its value is dependent on the value of *x*. If the ordered pair formed by a mapping
is not unique, the mathematical relationship that formed them is not a function; for example, the relationship **y ^{2} = x** is not a function because there is more than one value of y for each value of x.

The relationship

A function can also be expressed without explicitly indicating the dependent variable.
For example, the function **y = 2x ^{2} + x + 4** can be expressed as

A function can have as many independent variables as possible. However, keeping track of many independent variables that are represented by just as many letters can be tedious. So
mathematicians use a single letter with a variable integer subscript to represent the independent variables in a multivariate function. For example, suppose a function has five independent variables,
instead of using five different letters to represent these variables, a single letter, say x, is used as follows: x_{1}, x_{2}, x_{3}, x_{4}, x_{5}. The xs are
different independent variables because their subscripts are different. So the function is indicated as f( x_{1}, x_{2}, x_{3}, x_{4}, x_{5}).

Suppose we are able to reverse the mapping of the domain X onto Y. In other words, suppose
the function f(x) exists and there is a relation that matches the range of the function *f* with
the domain of *f*, to form the unique ordered pair (y, x), where y is the independent variable and x is the dependent variable. Such a relationship is called the **inverse function**
of *f* and it is indicated as f^{-1}(y).

The *argument* of a function *f* can be another function *g*. When this is the case, *f* is called a **composite function**. For example, the independent variable,
x of the function f, can itself be a function of another independent variable t, that is, x =g(t). Consequently, the function f(x) becomes the composite function f(g(t)). There is no limit to the number of functions that can be nested or
contained in another function. Composite functions are evaluated from the inside out, that is, the innermost function is the first to be evaluated, then the next innermost function and so on. The composite function f(g(t)) can also be expressed
as follows: **f o g: T mapped to Y**. In this expression, the function g receives its independent variable from the domain T, the result is then received as the independent variable
for the function f which then produces an output in the range Y. An intermediary domain (X in this example) which contains the output of the function g is implied in this representation of a composite function. In general, the number
of intermediary domains is equal to the number of nested functions in the composite function.

The evaluation of some functions for a given domain requires
the recursive inputs of the results of prior evaluations of the function. Such functions are called
**recursive functions**. A useful recursive function normally has an initialization condition and a termination condition.
A recursive function without a termination condition is considered an indefinite iteration in computer programming.
Consider the following function:

** = 1!** when n =1

**f(n)**

** = n(f(n-1))** when n an integer > 1

The symbol "!" is called *factorial* in mathematics. When it follows a number, it means the multiplication of all positive intergers equal to and less than the number.
For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. Now from the definition of the function f(n), we have f(1) = 1! = 1, f(2) = 2(f(2-1)) = 2(f(1)) = 2 x 1! = 2 x 1 = 2!.
In order to get the value of the function when n=2, we needed the value of the function when n = 1. This is a fundamental characteristic of recursive functions.

Suppose the domains and codomains of f and g are the same, that is, function f maps X to R and the function g maps X to R . Then the arithmetic operations on f and g for all x in X are defined as follows:

(f + g)x = f(x) + g(x)

(f - g)x = f(x) - g(x)

(f x g)x =( f(x) )(g(x))

(f/g)x = f(x)/g(x).

Peter Oye Sagay