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Unformatted text preview: Math021, week 1 Functions and their Graphs Definition 1.1 A function f is a rule assigning a number to each of the num- bers. The number assigned to the number x via the rule f is usually denoted by f ( x ) . Remark 1.2 The objective of our course “Calculus” is to study functions. Example 1.3 Let f be the rule of assignment so that for each number x , its square x 2 is assigned. The handy notation to define this function is to write f ( x ) = x 2 for all x . Example 1.4 Similarly, 1. g ( x ) = | x | for all x . 2. h ( x ) = 1 + x 4 for all x . are example of functions. However, 1. p ( x ) = 1 x 2. q ( x ) = √ x are somewhat different because they do not assign a number to EVERY choice of a number. Definition 1.5 The domain of a function f is the collection of numbers x such that a number is assigned to x via the rule f (we also say that f ( x ) is defined). Example 1.6 The domain of the function p ( x ) = 1 x is the collection of all numbers except . Example 1.7 The domain of the function q ( x ) = √ x is the collection of all non-negative numbers. Definition 1.8 Two coordinate-axes in the plane are two straight lines perpen- dicular to each other. One of them is usually called the “x-axis”, another line is usually called the “y-axis”. For every pointusually called the “y-axis”....
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- Fall '10