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Precalc0101to0102

# Precalc0101to0102 - CHAPTER 1 Functions 1.1 Functions 1.2...

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CHAPTER 1: Functions 1.1: Functions 1.2: Graphs of Functions 1.3: Basic Graphs and Symmetry 1.4: Transformations 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus 1.6: Combining Functions 1.7: Symmetry Revisited 1.8: x = fy () 1.9: Inverses of One-to-One Functions 1.10: Difference Quotients 1.11: Limits and Derivatives in Calculus • Functions are the building blocks of precalculus. • In this chapter, we will investigate the general theory of functions and their graphs. • We will study particular categories of functions in Chapters 2, 3, 4, and even 9.

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(Section 1.1: Functions) 1.1.1 SECTION 1.1: FUNCTIONS LEARNING OBJECTIVES • Understand what relations and functions are. • Recognize when a relation is also a function. • Accurately use function notation and terminology. • Know different ways to describe a function. • Find the domains and/or ranges of some functions. • Be able to evaluate functions. PART A: DISCUSSION WARNING 1 : The word “function” has different meanings in mathematics and in common usage. • Much of precalculus covers properties, graphs, and categorizations of functions. • A relation relates inputs to outputs . • A function is a relation that relates each input in its domain to exactly one output in its range . • We will investigate the anatomy of functions (a name such as f , a “function rule ,” a domain, and a range), look at examples of functions, find their domains and/or ranges, and evaluate them at an input by determining the resulting output .
(Section 1.1: Functions) 1.1.2 PART B: RELATIONS A relation is a set of ordered pairs of the form input, output () , where the input is related to (“yields”) the output . WARNING 2 : If a is related to b , then b may or may not be related to a . Example 1 (A Relation) Let the relation R = 1, 5 , ± ,5 , 7 {} . ± R , so 1 is related to 5 by R . Likewise, is related to 5, and 5 is related to 7. R can be represented by the arrow diagram below. § PART C: FUNCTIONS A relation is a function ± Each input is related to (“yields”) exactly one output. A function is typically denoted by a letter, most commonly f . Unless otherwise specified, we assume that f represents a function. The domain of a function f is the set of all inputs . It is the set of all first coordinates of the ordered pairs in f . We will denote this by Dom f , although this is not standard. The range of a function f is the set of all resulting outputs . It is the set of all second coordinates of the ordered pairs in f . We will denote this by Range f . • We assume both sets are nonempty. • (See Footnote 1 on terminology.)

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(Section 1.1: Functions) 1.1.3 Example 2 (A Relation that is a Function; Revisiting Example 1) Again, let the relation R = 1, 5 () , ± ,5 ,7 {} . Determine whether or not the relation is also a function . If it is a function, find its domain and its range .
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Precalc0101to0102 - CHAPTER 1 Functions 1.1 Functions 1.2...

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