Functions - Contents 1 Functions 21 1.1 Definition of a...

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Unformatted text preview: Contents 1 Functions 21 1.1 Definition of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2 Examples of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.1 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.2 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.4 The Absolute Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2.5 The Square Root Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2.6 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2.7 The Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2.8 Floor Function b x c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2.9 Ceiling Function d x e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2.10 One to One Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2.11 Even and Odd Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 CONTENTS Chapter 1 Functions 1.1 Definition of a Function A function , f , with domain D , is a rule which assigns to each element x ∈ D a single real number, f ( x ). The domain is usually a set of real numbers. The range of f consists of all numbers of the form f ( x ), with x ∈ D . The rule, f , can be thought of as a machine designed to take a specified set of real numbers (the domain) and produce, for each acceptable x of the input, a single real number f ( x ). f : D → B x A A A Domain A f B Range f ( x ) x 1 x 2 x 3 22 Functions 1.2 Examples of Functions 1.2.1 Polynomial Functions (i) Constant Functions: All functions of the form f : R → { a } where R is the set of real numbers and { a } is the set containing the fixed real number a . Examples : (1) f ( x ) = 2. Here f assigns to each number x ∈ R the real number 2. (2) Z ( x ) = 0. To every real number Z assigns the real number zero. This function is called the zero function, or zero polynomial. (ii) Linear Functions: All functions L : R → R defined by the rule L ( x ) = ax + b , where a and b are any fixed real numbers. Examples : (1) I ( x ) = x . I assigns every real number to itself. This function is called the identity function....
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This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.

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Functions - Contents 1 Functions 21 1.1 Definition of a...

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