# Ch 2: The Basics of Functions

### About This Chapter

## The Basics of Functions - Chapter Summary and Learning Objectives

When you figure out the cost of gas in relationship to the amount of gallons purchased, you are using mathematical functions to find your answer. In simplest terms, a function is really nothing more than a mathematical rule or a particular type of equation you use to solve everyday problems. There are many different aspects to functions, though, and the video lessons in this chapter let you explore all the ins and outs, so you can solve a full range of mathematical equations. This chapter is designed to teach you about:

- Function basics
- The difference between domain and range
- The rules for composing functions
- Inverse functions

Video | Objective |
---|---|

What is a Function: Basics and Key Terms | Define dependent and independent variables, analyze the domain of a function, and interpret the range of a function. |

Functions: Identification, Notation & Practice Problems | Describe a function as a kind of equation, go over function notation, and practice the vertical line test to verify functions. |

Transformations: How to Shift Graphs on a Plane | Break down what transformations do to functions, determine the effects that reflections and rotations have on functions, and then investigate the impact that translations and dilations have on functions. |

What Is Domain and Range in a Function? | Decipher the concepts of inputs and outputs, establish graphing rules, and then figure out how to graph the domain and range. |

How to Add, Subtract, Multiply and Divide Functions | Distinguish how to use substitution and simplification to perform these mathematical procedures with functions. |

How to Compose Functions | Discover how to replace one function for another when composing, learn the steps for working your way through equations, and identify how to check your work. |

Inverse Functions | Explain the meaning and use of the following expression: f^-1(x), and then review the steps for the horizontal line test. |

Applying Function Operations Practice Problems | Develop sample problems that combine most of the major function-related terms discussed in the previous lessons. |

### 1. What is a Function: Basics and Key Terms

Mapping numbers sounds complex, but we do it when we buy gasoline. We pump gasoline, and the gas station charges us based on the amount of gas that we pump. Learn how this relates to functions while reviewing the basics and notations in this lesson.

### 2. Functions: Identification, Notation & Practice Problems

A function is simply a rule that takes one number and turns it into another. But some special conditions must apply for it to be a true mathematical function. Learn about those conditions and how we write functions here!

### 3. Transformations: How to Shift Graphs on a Plane

What is a transformation? Well, it's something that transforms one function into another! To see what I mean and how that looks, check out this lesson!

### 4. What Is Domain and Range in a Function?

The domain and range are the possible outputs and inputs of a function. In this lesson, learn about what might restrict the domain and how to figure out the domain and range from a graph.

### 5. How to Add, Subtract, Multiply and Divide Functions

Adding, subtracting, multiplying and dividing functions is about as simple as substituting in expressions and then just doing whichever operation it asks you to do. Check out this video lesson to see some examples of this and learn just how easy it is!

### 6. How to Compose Functions

Function composition is the process of putting two or more functions together. This video lesson will explain how this process works and also show you how to evaluate functions that have been composed.

### 7. Inverse Functions

Inverse functions are two functions that do exactly opposite things. Check out this lesson to learn about how to write inverse functions, find inverse functions, and predict whether or not they exist.

### 8. Applying Function Operations Practice Problems

In this lesson, learn how to apply all the different properties of functions to solve complex problems. From function operations and composition to domain and range, get your practice here!

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### Other Chapters

Other chapters within the AP Calculus AB: Exam Prep course

- Graph Basics
- How to Graph Functions
- Limits of Functions
- Continuity of Functions
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Properties of Derivatives
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Using Derivatives
- Computing Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Understanding & Applying Integration Techniques
- Approximating Definite Integrals
- Using a Scientific Calculator for Calculus
- About the AP Calculus Exam
- AP Calculus AB Flashcards