Chapter 1: Functions and Models

Functions and Models explains function notation, domain, range, and the concept of piecewise-defined functions. Common families of functions—linear, polynomial, rational, trigonometric, exponential, and logarithmic—are reviewed in detail, with discussion of their properties, transformations, and real-world modeling applications. Graphical concepts such as symmetry, intercepts, and periodicity are explored to prepare students for interpreting and sketching functions. The chapter highlights the use of mathematical modeling to describe phenomena in physics, biology, economics, and engineering, introducing the idea of fitting functions to data. Students are guided through composite functions, inverse functions, and the importance of function composition in modeling multi-step processes. Trigonometric functions are revisited with an emphasis on their radian measure, unit circle definitions, and periodic behavior. Exponential growth and decay models are introduced, along with logarithmic scaling and change of base formulas. By the end of the chapter, readers are equipped to identify, analyze, and manipulate functions in preparation for the core concepts of calculus—limits, derivatives, and integrals.