# Ch 7: Functions and Graphs

### About This Chapter

## Functions and Graphs - Chapter Summary

If you need a refresher on mathematical functions and graphs, this chapter's got you covered. Lessons begin with tips for identifying functions and writing them with the correct notation. You can learn how to use the vertical line test to determine whether or not a graph of an equation is a function and review rules for finding the domain and range. Instructors also cover steps for identifying even, odd and one-to-one functions and writing and graphing an inverse function.

Methods for performing the rotation, dilation, translation and reflection transformations used to shift graphs on a plane are discussed in the second half of the chapter, as are tips for adding and subtracting functions. You can also learn how to perform the correct order of operations or use the quotient of powers to multiply and divide them. The chapter concludes with instruction on composing functions and using graphs to conjecture identities. Video lessons are designed to cover the following main topics:

- Types of functions
- Function domain and range
- Transformation of functions
- Function operations

These lessons include examples and practice problems to help you master the steps required to solve functional equations. Content is available in the form of short, engaging videos and transcripts that include links to supplemental text lessons, which you can use to review some of the mathematical terminology you're not quite familiar with. There are also self-assessment quizzes that allow you to practice solving problems.

### 1. Relation in Math: Definition & Examples

In this lesson, you will learn the definition of relation in terms of mathematics, as well as the various ways of displaying relations. We will also look at some examples.

### 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. 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.

### 4. Describe the Functional Relationship Between Quantities

In this lesson, you'll learn how to look at a graph, table, or function written mathematically and verbally describe the functional relationship shown. For example, you'll learn to describe whether the function is increasing or decreasing.

### 5. One-to-One Functions: Definitions and Examples

Watch this video lesson to learn what makes a one-to-one function different from a regular function. Learn a simple test you can use to check whether a function is one-to-one or not.

### 6. Understanding and Graphing the Inverse Function

If you use a function to map a to b, is there a way to go back from b to a again? Learn how to find and graph inverse functions so that you can turn a into b and back into a.

### 7. Comparing Function Transformations

Transformations of functions are useful in the analysis of functions. This lesson explains how transformations compare for different types of functions. Then, we will look at transformations of exponential functions, specifically.

### 8. 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!

### 9. 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.

### 10. 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.

### 11. Graphing Basic Functions

Graphs are just like maps - when you know the language! Review how locations have x and y coordinates similar to latitude and longitude, and how to plot points in the Cartesian plane.

### 12. Using Graphing Technologies to Graph Functions

Math, in some ways, is a lot easier to do today than it was a hundred years ago. In this lesson, you'll learn about the various technologies you can use to help you graph your functions, even 3-D ones.

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### Other Chapters

Other chapters within the TExES Mathematics 7-12 (235): Practice & Study Guide course

- About the TExES Math 7-12 Exam
- Real Numbers
- Mathematical Models
- Complex Numbers & the Complex Plane
- Number Theory
- Number Patterns
- Linear Functions
- Quadratic Functions & Polynomials
- Evaluating Piecewise & Composite Functions
- Rational and Radical Functions
- Inequalities and Absolute Values
- Exponentials & Logs
- The Unit Circle
- Trigonometric Functions
- Using a Scientific Calculator for Calculus
- Understanding Limits in Math
- Understanding Rate of Change
- Calculating Derivatives of Functions
- Derivatives and Graphs
- Optimization in Calculus
- Definite Integrals and Sums
- Integration Applications in Calculus
- Working with Measurement
- Finding Volume, Area & Perimeter
- Introduction to Proofs and Constructions
- Congruence and Similarity
- Triangles
- Circles
- Real World Shapes
- Coordinate Geometry
- Understanding Transformations in Math
- Conic Sections
- Understanding Vectors
- Measuring & Displaying Data
- Data Distribution Overview
- Sampling in Statistics
- Distribution & Inference in Statistics
- Inference About a Mean
- Regression and Correlation
- Finding Probability
- Probability Distributions and Statistical Inference
- Experiments and Surveys
- Mathematical Process & Perspectives
- Teaching Strategies & Activities for the Math Classroom
- Differentiated Instructional Strategies for the Math Classroom
- Using Student Assessments in the Math Classroom
- TExES Mathematics 7-12 Flashcards