All about Functions: Mathematics Primer with 7000 Solved

Functional Analysis is Very Important

Functions form an important branch of mathematics: being a core concept in precalculus at the middle and high school levels, and an important part of functional analysis at more advanced levels.

Important concepts

The concept of a function, at an introductory level, is incredibly simple, but leads to very profound consequences. The elementary concept of a black box that maps one set of numbers to another set of numbers may seem trivial, but within a few short years of being introduced to this concept, high school students can use it to explore the concepts of limits, continuity, and differentiability of functions, and to lay a solid foundation for their exploration of differential and integral calculus.

Pedagogy followed

This book aims to make the introductory part of any first course in functions much easier and more rigorous to follow by providing detailed theory, solved examples, and practice exercise drills. This is the first part of my series on functions and functional analysis. To make sure that you can master the concepts at a basic level, I have concentrated on familiar functional forms in this book: starting with linear and then higher order polynomial (mainly quadratic) functions, I then devote some time to discussing the important concept of the inverse of a function, and then linking everything in this book together by showing you how to find the inverse of a polynomial function. I also introduce the concept of the zero(s) of a function - you might be familiar with this from your work on linear equations and quadratic equations.

How to use this book

You can use this book either as a quick introductory or refresher course in the field of functions. Within a 10-20 hour period, you should be able to cover the theory and the solved examples, to give yourself everything you need to move forward and use elementary functions in your work. However, the recommended way to use this book is to complete the drill worksheets as well; if you spend 50-100 hours going through every nook and cranny that this book contains, your foundation will be so strong that you will never have any fundamental doubts related to functions, relations, and mappings, ever again.

Looking beyond

Subsequent books in this series will focus on trigonometric and exponential functions, and on limits, continuity, and differentiability. In the end, I will also look at multivariate functions in two variables, x and y, or even more!