UNIT 1
REAL NUMBERS AND FUNCTIONS
Structure
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Introduction Objectives
Basic Properties of R Absolute Value Intervals on the Real Line Functions 1.5.1 Definition and Examples 1.5.2 Inverse Functions 1.5.3 Graphs o f Inverse Functions
New Functions from Old 1.6.1 Operations on Functions 1.6.2 Composite of Functions
Types of Functions 1.7.1 Even and Odd Functions 1.7.2 Monotone Functions 1.7.3 Periodic Functions
summary Solutions and Answers
1.1
INTRODUCTION
This is the first unit of the course on Calculus. We thought it would be a good idea to acquaint you with some basic results about the real number system and functions, before you actually start your study of Calculus. Perhaps, you are already familiar with these results. But, a quick look through the pages will help you in refieshing your memory, and you will be ready to tackle the course.
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In the next three sections of this unit, we shall present some results about the real number system. You will find a number of examples of various types of functions in Sections 1.5 to 1.7. You should also study the graphs of these functions carefully, in order to be able to vistlalise given functions. In fact, try to draw a graph whenever you encounter a new function. We shall , systematically study the tracing of c w e s in Block 2 Unit 4.
Objectives After reading this unit you should be able to : "3
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recall the basic properties of real numbers derive other properties with the help of the basic ones ichtify various types of bounded and unbounded intervals define a function and examine whether a given function is one-onelonto investigate whether a given function has an inverse or not deiine the scalar multiple, absolute value, sum, difference, product, quotien't of the given functions and determine wl$&er a given function is even odd, monotonic or periodic.
1.2
BASIC PROPERTIES OF R
In the next three sections, we are going to tell you about the set R of real numbers, which is allpervading in mathematics. The real number system is the foundation on which a large part of mathematics, including calculus, rests. Thus,before we actually start leaming calculus, it' is necessary to understand the structure of the real number system. You are already familiar with the operations of addition, subtraction, multiplication and division of real numbers, and also withGkqualities.Here we shall quickly recall some of their properties. We start with the operation of addition: