Advanced Mathematics Self-study and Problem Solving Guide (Second Edition)

Author: None
Publisher:
Publish Date: 1999-12-01
Features: This self-study guide has been revised to accompany the "Advanced Mathematics" textbook (Second Edition) compiled by the Correspondence Mathematics Teaching and Research Office of Tongji University. It summarizes the basic concepts and theorems of each chapter, analyzes problem-solving approaches, and compares different methods. The book provides a rich and typical set of examples, with detailed solutions and key steps to focus on as well as common mistakes to avoid, helping readers deepen their understanding of the course, broaden their thinking, and reduce difficulties in self-study and problem-solving. This book can serve as a supplementary teaching material for adult education in various engineering fields, as well as a reference for engineering students and technicians studying advanced mathematics independently.
Excerpt: Determining the domain of a function means finding the range of values for the independent variable that make the function meaningful within the real number system. The key to solving problems correctly lies in:
(1) Clarifying the domains of the five types of basic elementary functions, such as even-root functions requiring non-negative radicands; logarithmic functions requiring arguments greater than zero; and the domains of inverse trigonometric sine and cosine functions being the interval [?1, 1], etc.
(2) Understanding that the denominator of a rational function cannot be zero.
(3) Mastering the methods of solving inequalities (sets) (it is necessary to review relevant knowledge from secondary school).
(4) For practical problems, considering whether the problem itself is meaningful.
II. Inverse Functions, Composite Functions, and Elementary Functions
Definition of an Inverse Function
Let the domain of the function \( y = f(x) \) be the set of real numbers \( D \) and its range be the set of real numbers \( W \). If for every \( y \in W \), there is a unique \( x \in D \) corresponding to it through the relation \( y = f(x) \), then the function \( x = \psi(y) \) thus determined is called the inverse function of \( y = f(x) \). The original function \( y = f(x) \) is referred to as the direct function. For the inverse function \( x = \psi(y) \), it is customary to write it as \( y = \psi(x) \).
Composite Functions
If two functions \( y = f(u) \) and \( u = \psi(x) \) are given, and the range of \( u = \psi(x) \) is entirely or partially contained within the domain of \( y = f(u) \), then through the connection of \( u \), \( y \) is also a function of \( x \). This function is called a composite function formed by the functions \( y = f(u) \) and \( u = \psi(x) \), denoted as \( y = f[\psi(x)] \), where \( u \) is called the intermediate variable.