# Example 044 if x 2 0 2 and y 5 0 1 use the triangle

• Notes
• 228
• 100% (4) 4 out of 4 people found this document helpful

This preview shows page 19 - 23 out of 228 pages.

##### We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
The document you are viewing contains questions related to this textbook.
Chapter 6 / Exercise 6
Calculus
Stewart
Expert Verified
11 Example 0.4.4 If | x - 2 | < 0 . 2 and | y - 5 | < 0 . 1 , use the triangle inequality to estimate | ( x + y ) - 7 | .
Example 0.5.1 Find all the read numbers x R such that the following is true, | 2 x - 4 | ≥ 5 .
##### We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
The document you are viewing contains questions related to this textbook.
Chapter 6 / Exercise 6
Calculus
Stewart
Expert Verified
12 Chapter 0. Introduction
CHAPTER ONE Functions This section will focus on what is a function, different functions and how they can be used. 1.1 FOUR WAYS TO REPRESENT A FUNCTION See section 1.1 of the textbook In high school mathematics we saw linear functions (also called linear mappings). We also saw quadratic functions (mappings) as well as a bunch of other examples. Here, we take a step back and define very precisely what we mean by a function. Then we will build up our understanding of them by developing a rather extensive toolbox. From the textbook we have the following definition. Definition 1.1.1. A function f is a rule that assigns to each element x in a set D exactly one element, called f ( x ) , in a set E . This can be described as, f : D E This says that f is a function, or a mapping , from the set D to the set E . In this course we will assume that both D and E are sets of real numbers. We can describe f as a real-valued function of a single real variable. There are many different interpretations of functions. The function f can be seen as a machine that takes an input, x , and spits out an output, f ( x ) . Can picture a function as an arrow diagram that maps every point in the set D to a unique point in the set E . 13
14 Chapter 1. Functions Example 1.1.1 If f ( x ) = x 2 + 1 and h = 0 compute, f ( a + h ) - f ( a ) h . Solution. We substitute not the expression and simplify, f ( a + h ) - f ( a ) h = ( a + h ) 2 + 1 - ( a 2 + 1) h , = a 2 + 2 ah + h 2 - a 2 h , = 2 ah + h 2 h , = 2 a + h. J The graph of the function f is the set of ordered pais, { ( x, f ( x )) | x D } . When discussing functions two important ideas are the domain and the range. Definition 1.1.2. The domain of the function f is all the points x R where f ( x ) is defined. The domain can be the whole real line but it need not be. That is to say functions need not be define everywhere, for example 1 /x . Definition 1.1.3. The range of the function is the set of all values f ( x ) where it is assumed that x is in the domain of the function. These ideas extend nicely to higher dimensions, as you will see in subsequent courses.