Functions

This chapter introduces the language the rest of the book is written in. It begins with what makes a relation a function and how to read one from a formula, a table, or a graph, then builds the tools for describing how a function behaves: its domain and range, how fast it changes, where it rises and falls, and where it reaches its highest and lowest values. It closes by building new functions from old ones — combining and composing them, shifting, reflecting, stretching, and compressing their graphs, and running them backwards to form inverses.

Sections

• Functions and Function Notation — deciding whether a relation is a function, function notation, evaluating and solving functions given by formulas, tables, and graphs, one-to-one functions, the vertical and horizontal line tests, and the toolkit functions.
• Domain and Range — finding the domain of a function from its equation, describing sets with inequality, set-builder, and interval notation, reading domain and range from a graph, the domains and ranges of the toolkit functions, and graphing piecewise-defined functions.
• Rates of Change and Behavior of Graphs — average rate of change from tables, graphs, and formulas, the intervals on which a function increases or decreases, and locating local and absolute maxima and minima.
• Composition of Functions — combining functions with sums, differences, products, and quotients, building and evaluating composite functions from tables, graphs, and formulas, finding the domain of a composition, and decomposing a function.
• Transformation of Functions — vertical and horizontal shifts, reflections about the axes, even and odd functions, vertical and horizontal stretches and compressions, and performing a sequence of transformations.
• Absolute Value Functions — absolute value as distance, graphing absolute value functions and their transformations, and solving absolute value equations and inequalities algebraically and graphically.
• Inverse Functions — verifying inverse functions, the domain and range of an inverse, restricting a domain to make a function one-to-one, evaluating an inverse from tables and graphs, finding an inverse from a formula, and reflection about the line $y=x$.

This chapter is adapted from Precalculus 2e, Chapter 1: Functions by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Each section page records its own adaptations.