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

Author: Tongji University Press
Publisher:
Publish Date: 1999-12-01
Features: This self-study guide has been revised to complement the textbook "Advanced Mathematics" (Second Edition) compiled by the Department of Mathematics for Correspondence Education at Tongji University. The book summarizes the basic concepts and theorems of each chapter, analyzes problem-solving approaches, and compares different methods of solving problems. The examples provided are rich and typical, with detailed solutions and hints on key steps and common mistakes to help readers deepen their understanding of the course, broaden their thinking, and reduce difficulties in self-study and problem-solving. This book can be used as a supplementary textbook for adult education in engineering disciplines or as a reference for engineering students and technicians studying advanced mathematics.
Excerpt: Determining the domain of a function means finding the range of values of the independent variable that make the function meaningful within the real number system. The key to solving problems correctly is: (1) understanding the domains of the five types of basic elementary functions, such as even-root functions where the radicand must be non-negative; logarithmic functions where the argument must be greater than zero; and the domains of inverse trigonometric sine and cosine functions being the interval [－1, 1], etc.; (2) clarifying 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: Suppose the domain of the function \( y = f(x) \) is the set of real numbers \( D \) and its range is the set of real numbers \( W \). If for every \( y \in W \), there is a unique \( x \in D \) corresponding to \( y \) through the relationship \( y = f(x) \), 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.