Functions
By David493
Table of Contents
I. Introduction
II. Definitions
III. Examples
I. Introduction
In this guide, I will discuss the mathematical use of the word function. The concept of a function is very basic in mathematics. In general terms the word function is used to denote a specific association between the elements of two sets. Thus, in order to have a function we shall need two sets, which we shall order as a first and second set, and then a rule that pairs a member of the second set with each member of the first set.
II. Definitions
Let X be the first set and Y be the second set.
A function f, is an assignment by some rule of a unique (one and only one) element of Y to each element of X
The set X is called domain of f and each element x ∈ X is called an argument of f.
The set Y is called the range of f and each element y ∈ Y is called a value of f.
Since for each element x ∈ X a unique element of y ∈ Y is determined, we are forming a set of ordered pairs (x,y) in which to each x there is associated only one y. Thus a function of f from X to Y is this set of ordered pairs.
The rule which assigns an element y to a specified argument x may take many forms. The most frequent form is that of an equation. The rule can also be a simple listing of the ordered pairs, or a graph, or a statement.
The notion for a function also takes many forms. The most frequent form in advanced mathematics is f: X to Y, which is read "the function f from X to Y." If we want to emphasize the elements of the sets, we write f : x to ywhich is read "f takes x into y" The elements of the range are donated by f(x), which is read "the value of f at x"* This notation permits the simple indication of the value of the function for a sepcified x; e.g., f(2) means the value (y) at x= 2.
Since a function from X to Y is a set of ordered pairs (x, y) with x ∈ X and y to Y we may use set notation to describe a function:
* The notion f(x) was also used for many years to denote the function itself and was read "f of x" This emphasized that we were discussing a function of the single argument x.
In our discussion of a function we have called the elements of the domain, x ∈ X, arguments of the function. Whenever the set of X has more than one element, the elements are also variables and they may be arbitrarily selected. Consequently, the elements of the domain are also called independent variables of the function. On the other hand, the elements of the range, y ∈ Y, depending upon the choice of x for their values. Thus, these elements are also called the dependent variables.
We have been using the letter f to represent a function. Other letters can be used, of course, and when more than one function is discussed at a time, we frequently use letters like g,h,ietc.
III. Examples
Due to the importance of the concept of function we shall devote this section to a discussion of examples of functions
Example 1. Let the function f be given by the equation
and let the domain be set of real numbers. We can now tabulate as many ordered pairs as we please by giving values to x and calculating the corresponding values of y. Some of the ordered pairs are (0, 1), (1,-5), (-1,11), (2, -7), (3, - 5).
Example 2. Consider a table of values
This table of values relates each value of x to a unique value y and thus establishes the set of ordered pairs (1,5), (2,7), (3,9), (4,11). The table defines a function whose domain is X = {1,2,3,4} and whose range is Y = {5,7,9,11}.
Example 3. The area of a circle of radius r is given by formula A = πr². The number π is a fixed value, a constant; thus the area A depends upon r for its value, and we can write A(r) = πr². Since we are considering a circle, the radius is a non negative number, r ≥ 0. The formula defines a function whose domain is the set of non negative real numbers and whose range is also the set of non negative real numbers.
Example 4. Consider all the cities in the state of Ohio. Each city is marked on a map of Ohio. The rule that associates a name with position on the map is the function that takes X = {names of cities} to Y = {map positions}.
Example 5. The multiplicative-inverse axiom for real numbers says that there is a unique inverse 1/x for each x ∈ R except x = 0. This can be expressed as the function f: x to 1/x, whose domain and range is the set of real numbers R providing x ≠ 0.
Example 6. Consider the function f defined by
In this case two equations are required to define the function, depending upon the interval in which the elements of the domain are located. The domain in all non negative real numbers. The calculation of the range becomes an important operation and, as a general problem, will be discussed throughout the book. In this example we can calculate the following set of ordered pairs:
and notice that the values of y seem to fall between 2 and 5. The range is in fact [2,5]. One method for verifying the range is to "graph the function,"a concept we shall discuss in the next sections. this example also illustrates a special function. for all values of x > 3,the values of the function becomes the same fixed number,5. Such a function is called a constant function and is written y = c or f: x to c where it is assumed that c is a constant.
Comments are welcome.
Thanks for this guide, and on the criticism note... I like the colors, pictures and good content. But, you need to check your grammar in some areas. That's just me being picky though.
