Chapter_minus1 - Chapter -1 Mathematical Refresher The...

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© Sanjay K. Chugh 5 Spring 2008 Chapter -1 Mathematical Refresher The concept of a function is a very general and powerful one. A function is a mathematical object that serves as a fundamental tool in many fields of analysis. We will not here give a rigorous or comprehensive treatment of the mathematical notion of a function. The purpose here is to (re)familiarize you with the basic concepts and the most important ways in which we will use functions as we develop tools of economic analysis in this course. Abstract Functions and Functional Forms A function transforms an input into an output. More specifically, a function is a rule that specifies how an input is to be transformed into some output. At its simplest level, the level with which we will be concerned, the input and outputs will all be numbers. In general, any function can have multiple inputs and multiple outputs. Every function that we will use will have only one output – that is, a function whose operation results in only one numeric value as its output. However, we will regularly encounter functions that have multiple inputs, in addition to functions that have simply a single input. A function can be written and used in abstract form, as when we simply write and use the function () f x without specifying anything further about what the function actually does. Often, however, in order to do something useful with a function, we need to specify a particular functional form – that is, we often need to specify what a function actually does (i.e., what the rule is). Some examples of common functional forms will help illustrate the concept: 2 f xx ( 1 . 1 ) () 2 8 f ± ( 1 . 2 ) f ( 1 . 3 ) () l n f ( 1 . 4 ) (, ) l n () 0 . 8 l n f xy x y ± ( 1 . 5 ) In the above simple functions (functional forms), note that each function returns only one number as its output (as promised). Also note that function (1.5) is a function of two inputs, while the others are all functions of one input. Arguments of Functions To be a bit more formal mathematically, an input(s) to a function is commonly known as its argument(s), and the output of a function is commonly known as its result or value.
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© Sanjay K. Chugh 6 Spring 2008 Using the functions defined above, we see that each of the functions (1.1) through (1.4) takes one argument named x . Function (1.5) takes two arguments named x and y . When actually performing numerical calculations using functions, the x in each case of the functions (1.1) through (1.4) would be replaced with an actual number because it is meaningless to square the letter x, because only numbers can be squared. The leads to the distinction between formal arguments and actual arguments. Think of a formal argument as a placeholder in an abstract function. In each function (1.1) through (1.4), the formal argument is x . In function (1.5), the two formal arguments are x and y . More will be said about replacing formal arguments with actual arguments, but first let’s examine the components of a function definition.
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Chapter_minus1 - Chapter -1 Mathematical Refresher The...

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