This textbook focuses on the study of different kinds of elementary functions ubiquitous both in high school Algebra and Calculus. To analyze the functions ranging from polynomial to trigonometric ones it uses rudimentary techniques available to high school students and at the same time follows the mathematical rigor appropriate for university level courses.Contrary to other books of PreCalculus this textbook emphasizes the study of elementary functions with rigor appropriate for university level courses in mathematics although the exposition is confined to the prelimit topics and techniques. This makes the book useful on the one hand as an introduction to mathematical reasoning and methods of proofs in mathematical analysis and on the other hand as a preparatory course on the properties of different kinds of elementary functions.The textbook is aimed at university freshmen and highschool students interested in learning strict mathematical reasoning and in preparing a solid base for subsequent study of elementary functions at advanced level of Calculus and Analysis.