Mathematical Functions(2)

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 y2 = x is not a function because there is more than one value of y for each value of x.
The relationship y = 2x2 + x + 4 is a function. Three ordered pairs associated with this function are (0,4), (1,7), (2,14). The independent variable x, is assigned the values 0, 1 and 2. Each of these values called the argument of the function, are then subsituted in the function to get the respective value 4, 7 and 14 of the dependent variable.

A function can also be expressed without explicitly indicating the dependent variable. For example, the function y = 2x2 + x + 4 can be expressed as f(x) = 2x2 + x + 4. In this format "y" is replaced by "f(x)", a notation that is interpreted as "function of x". Consequently, f(0) means the value of the function when x = 0, f(1) means the value of the function when x = 1, and so on. This format is used often when there are more than one independent variables. For example, the function f(x, y) = 2x2y + x y + 4 has two independent variables. This function can also be expressed as z = 2x2y + xy + 4, where z is the variable that is dependent on the values of x and y.

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: x1, x2, x3, x4, x5. The xs are different independent variables because their subscripts are different. So the function is indicated as f( x1, x2, x3, x4, x5).

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).

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Peter Oye Sagay

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