Are sequences functions? What can't the popular "vertical line test" be applied in some cases to determine if a relation is a function? How does the idea of rate of change connect with simpler ideas about proportionality as well as more advanced topics in calculus? Helping high school students develop a robust understanding of functions requires that teachers understand mathematics deeply. But what does that mean? This book focuses on essential knowledge for teachers about functions. It is organized around five big ideas, supported by multiple smaller, interconnected ideas--essential understandings. Taking teachers beyond a simple introduction to functions, this book will broaden and deepen their mathematical understanding of one of the most challenging topics for students--and themselves. It will help teachers engage their students, anticipate their perplexities, avoid pitfalls, and dispel misconceptions. They will also learn to develop appropriate tasks, techniques, and tools for assessing students' understanding of the topic. This book contains three chapters: (1) Functions: The Big Ideas and Essential Understandings; (2) Connections: Looking Back and Ahead in Learning; and (3) Challenges: Learning, Teaching, and Assessing. A foreword, a preface, an introduction, and a list of references are also included.