MA1102 Calculus Tutorial 2

Topics Covered:

Function. Its domain, codomain and range. Injective, surjective and bijective functions. Inequalities and simple manipulation of inequalites. The modulus function as 'distance' function.

Textbook: Chapter 2 and 3

How can we describe the range of a function? Remember that set notation here is very important. Stick to the same given notation. Do not invent your own, at least not just yet. Sum, difference, product and quotient of functions are easy notions. For composition of two functions, you will need to know when they can be composed. There are some deeper foundation questions not discussed. One is the very definition of a function without defining what a 'rule' is. This is not addressed in this tutorial. The other much more central question is this: What constitute a real number? Some of the partial answers can be found in

J.A. Green, Sets and Groups,

K. G. Binnmore, Sets, Numbers and Logic,

Ng Tze beng, Real Numbers ? (Not published yet)

Inequality is tied up with the notion of a total (linear) ordering on the real numbers R. The two axioms for a positive cone determine the only ordering on the real numbers; and from these two axioms spring forth the properties of inequaliy given in the lectures. Learn to use these properties effectively. The modulus function is introduced as a 'distance' function on the real line. There are some unanswered questions here that we hope you will discover the answer yourself sometime, some day. How can we understand the real numbers as much as and as well as we think we understand the rational numbers? For instance, what is Ö2 ? If we do not know what this is, how and where can we place it in the real number system or real line with respect to the ordering? We can sum this up in one statement: the real number is the only complete totally ordered field (upto isomorphism). You should from time to time extend your understanding of the real numbers step by step by going through the reference given. For this tutorial, assume the existence and properties of R and proceed.