s1_2 - 2. INTRODUCTION TO FUNCTIONS 15 2. Introduction to...

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2. INTRODUCTION TO FUNCTIONS 15 2. Introduction to Functions 2.1. Function. Definition 2.1.1 . Let A and B be sets. A function f : A B is a rule which assigns to every element in A exactly one element in B . If f assigns a A to the element b B , then we write f ( a ) = b, and we call b the image or value of f at a . Discussion This is the familiar definition of a function f from a set A to a set B as a rule that assigns each element of A to exactly one element B . This is probably quite familiar to you from your courses in algebra and calculus. In the context of those subjects, the sets A and B are usually subsets of real numbers R , and the rule usually refers to some concatenation of algebraic or transcendental operations which, when applied to a number in the set A , give a number in the set B . For example, we may define a function f : R R by the formula (rule) f ( x ) = 1 + sin x . We can then compute values of f – for example, f (0) = 1, f ( π/ 2) = 2, f (3 π/ 2) = 0, f (1) = 1 . 357 (approximately) – using knowledge of the sine function at special values of x and/or a calculator. Sometimes the rule may vary depending on which part of the set A the element x belongs. For example, the absolute value function f : R R is defined by f ( x ) = | x | = ± x, if x 0 , - x, if x < 0 . The rule that defines a function, however, need not be given by a formula of such as those above. For example, the rule that assigns to each resident of the state of Florida his or her last name defines a function from the set of Florida residents to the set of all possible names. There is certainly nothing formulaic about the rule that defines this function. At the extreme we could randomly assign everyone in this
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s1_2 - 2. INTRODUCTION TO FUNCTIONS 15 2. Introduction to...

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