Back To Course

STAAR Mathematics - Grade 8: Test Prep & Practice11 chapters | 67 lessons | 3 flashcard sets

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Log in here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Functions are an extremely important part of mathematics. Let's look at some different ways of representing a function and how to determine if a given representation is, in fact, a function.

Think back to the last time you ate at a restaurant, and try to recall the dessert menu. It probably looked something like this:

See how the price of the dessert is determined by the type of dessert? In other words, if I tell you the type of dessert I want, you can determine the price. This relationship is an example of a function. In a **function**, one variable is determined by the other. We take an input, plug it into the function, and the function determines the output. For example, if I told you I wanted tapioca pudding, you put it into the dessert menu function, and tell me that it costs $3.

Functions have many representations. In the previous example, we described the function in words, and the image of the dessert menu described the function using a mapping, which related the type of dessert to the price. We can determine that this is a function by making sure that a type of dessert doesn't map to two different prices. If this was the case, we couldn't determine the price from the type of dessert. For example, consider this dessert menu:

If I told you I wanted tapioca pudding, you couldn't tell me if it cost $2 or $3, so this wouldn't represent a function.

In general, a relationship is a function if for every input, there is exactly one output. When this is the case, we can determine the output based on the input. Let's consider a few more representations of functions and how to identify a function from these representations.

A function can be represented using ordered pairs. We simply write the inputs as the first coordinates and the outputs as the second coordinates. Consider our dessert menu example. We would represent this using ordered pairs like this:

By our function rule, no input can have more than one output, so a set of ordered pairs is a function as long as no two ordered pairs have the same first coordinate with different second coordinates. This is illustrated here:

The first set of ordered pairs is a function, because no two ordered pairs have the same first coordinates with different second coordinates. The second example is not a function, because it contains the ordered pairs (1,2) and (1,5). These have the same first coordinate and different second coordinates.

We can use tables to represent functions by listing the input values in one column and the corresponding output values in another column. Let's look at our dessert function using a table.

To determine if a table represents a function, we think back to our rule. No input can have more than one output. Thus, in our table, we can't have two entries with the same input and different outputs. Consider these tables:

The first table represents a function since there are no entries with the same input and different outputs. The second table is not a function, because two entries that have 4 as their input, but one has 7 as the output, and the other has 14 as the output.

We can also represent functions using graphs by plotting all the ordered pairs of a function on a coordinate axis. For example, consider the function *y* = 2*x* + 1. We graph this by graphing all the ordered pairs that satisfy the equation.

But how do we know this is a function? We have a really neat way of determining if a graph represents a function. A function can't have two ordered pairs with the same first coordinate and different second coordinates. Let's think about what would happen if two such points were on a graph.

Notice that if there are two points with the same first coordinate and different second coordinates, we can draw a vertical line through them. Since a graph that is a function can't contain two points like this, it makes sense that a vertical line drawn anywhere on the graph should only intersect the graph once. This is called the **vertical line test**, and it states that a graph is a function if a vertical line drawn anywhere on that graph only intersects the graph once.

Consider the graph of our line *y* = 2*x* + 1. Notice that if we draw a vertical line anywhere on that graph, it will only intersect the graph once, so the graph represents a function. Now, consider this graph:

With this graph, we can draw a vertical line as shown, and it intersects the graph more than once, so this graph does not represent a function.

A **function** is a relationship in which one variable is determined by the other variable. In a function, each input has exactly one output, so if a relationship has an input that has more than one output, that relationship is not a function.

We can represent functions in many different ways. For instance, we can use words, mappings, ordered pairs, tables, and graphs. In each of these representations, we can determine if we have a function using our rule. No input can have more than one output. Thus, in ordered pairs, no two ordered pairs can have the same first coordinate and different second coordinates. In tables, no two entries can have the same input with two different outputs. In graphs, we can use the vertical line test by making sure that a vertical line drawn anywhere on the graph only intersects the graph once.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Log in here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
6 in chapter 4 of the course:

Back To Course

STAAR Mathematics - Grade 8: Test Prep & Practice11 chapters | 67 lessons | 3 flashcard sets

- Calculating Directly & Inversely Proportional Quantities
- Linear and Nonlinear Functions 5:56
- Graphing Non-Proportional Linear Relationships 6:09
- Direct and Inverse Variation Problems: Definition & Examples 6:27
- Data Mining: Function Properties from Derivatives 9:50
- Identifying Functions with Ordered Pairs, Tables & Graphs 5:44
- Describe the Functional Relationship Between Quantities 4:49
- Writing & Evaluating Algebraic Expressions for Two-Dimensional Geometric Figures 4:10
- Go to Functions & Proportionality

- SIE Exam Study Guide
- Indiana Real Estate Broker Exam Study Guide
- Grammar & Sentence Structure Lesson Plans
- Foundations of Science Lesson Plans
- Career, Life, & Technical Skills Lesson Plans
- Business Costs, Taxes & Inventory Valuations
- Using Math for Financial Analysis
- Assessments in Health Education Programs
- Governmental Health Regulations
- Understanding Health Education Programs
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- Curt Lemon in The Things They Carried
- Religion in Jane Eyre: Analysis & Examples
- Intrapreneurship in the Hospitality Industry
- Saadat Hasan Manto: Biography & Works
- Using File-Carving & Scalpel Techniques to Recover Android Device Data
- Functional Text Activities for Middle School
- A Midsummer Night's Dream Fairies Role & Analysis: Peaseblossom, Cobweb, Moth & Mustardseed
- Quiz & Worksheet - Assessing Nutritional & GI Status
- Quiz & Worksheet - Achievements of President Jackson
- Quiz & Worksheet - Agile Environments
- Quiz & Worksheet - State & Federal Rights in the Civil War
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- Formative Assessment in Schools | A Guide to Formative Assessment
- Science Lesson Plans

- Introduction to Statistics: Homework Help Resource
- CSET Business Subtest I (175): Practice & Study Guide
- Abnormal Psychology: Help and Review
- Nursing 101: Fundamentals of Nursing
- ILTS TAP - Test of Academic Proficiency (400): Practice & Study Guide
- Beginning Steps in Business Writing
- AEPA Middle Grades ELA: Types of Poetry
- Quiz & Worksheet - HR Policies & Procedures
- Quiz & Worksheet - Stages of Change
- Quiz & Worksheet - Slope-Intercept Form
- Quiz & Worksheet - Features & Types of Corporations
- Quiz & Worksheet - The Albany Plan of Union

- What is a Corrective Action Plan? - Definition, Procedures & Examples
- Lincoln-Douglas Debates Lesson Plan
- To Be, Or Not To Be: Quote Analysis
- What is On the MCAT?
- How to Improve Math Skills
- New Careers for Teachers
- What is On the MCAT?
- Global History & Geography Regents Exam Info
- What Are Common SAT Essay Topics?
- When Do You Apply for Community College?
- How to Learn English Online
- Anti-Bullying Survey Finds Teachers Lack the Support They Need

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject