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1.1-fcns - Chapter 1 Functions and Limits 1.1 Functions We...

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Chapter 1: Functions and Limits 1.1 Functions We will think of functions in this course as input-output machines: a number goes in, the machines does something to it, and it spits out another number. input output Since the input and output can take on different values they are variables . The value of the output is determined by the value of the input. In other words, if the value of the input is known there is only one possible value for the output. For this reason the input is called the independent variable and the output is called the dependent variable . Notations for Functions Although there are many ways to denote functions, we will restrict ourselves, primarily, to two different ways. Most of the other ways are mixtures of these two ways. Function notation: When using function notation, we label the process (in other words the box). When there is no other compelling name, we will often label it f or g . f Using this label, the output can then be expressed in terms of the inputs. For instance, if the input is 2, the output is labeled f (2). If the input is x the output is f ( x ). If the input is - x 2 + 3 then the output is f ( - x 2 + 3). 2 f (2) f x f ( x ) f 1
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- x 2 + 3 f ( - x 2 + 3) f This is generally the notation that we use when we are doing theory and don’t have a particular application in mind. Example: Consider the function f given by f ( x ) = πx 2 . Notice that this formula describes the function. In other words it describes how the output is obtained from the input; the input is squared and then multiplied by π .
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