Back To Course

Common Core Math Grade 8 - Functions: Standards5 chapters | 19 lessons

Are you a student or a teacher?

Start Your Free Trial To Continue Watching

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.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*John Sepanski*

John has taught 6th Grade Mathematics through Geometry and has a Master's degree in Education

In this lesson, we will explore algebraic and numeric properties and the evaluation of functions. We'll demonstrate function evaluations and the idea that every element of the domain has a unique element in the range.

Have you ever wondered how a gum ball machine works? When you place a quarter into a particular machine, it drops down 14 skittles or M&M's, etc. But, how does it know how to do that? Well, if you take apart a gum ball machine, you will find that there are a series of gears that turn based upon the coin actually being a quarter.

As the gears turn, they rotate a plastic tray that has a bin in it that will hold exactly 14 pieces of candy. When that bin rotates beneath an open slot in the candy jar, gravity does its job and 14 pieces drop into the slot. Then, the plastic tray continues to turn and drops the candy down the chute to the waiting hand of the buyer. Yum! Algebraic functions are like that.

When you evaluate a function for a given value, which we call domain, you obtain an output, which we call an element of the range. **Domain** is the set of input values of an algebraic equation. **Range** is the set of output values of an algebraic equation. In an algebraic function, there is exactly one input for every output. Let's look at an example:

*y* = 2*x* + 4

If you evaluate this expression for 1, 2, 3, 4, you get these outcomes:

*y* = 2(1) + 4 = 6

*y* = 2(2) + 4 = 8

*y* = 2(3) + 4 = 10

*y* = 2(4) + 4 = 12

Notice that for each input there is exactly one output. Let's try evaluating for 3 again. Let's make sure that the answer doesn't change. Wait a minute! Why would the answer ever change? Isn't an input domain = 3 always going to produce an output of range = 10? *y* = 2(3) + 4 = 6 + 4 = 10. 2(3) is always going to be 6, and 6 + 4 can never be anything but 10. For this reason, this algebraic expression is a function.

A **function** has exactly one distinct output value in the range for each input value of the domain. It's like true love! There is that special someone out there in the range that is simply perfect for the element of the range! When you have a function expression, *y* = 2*x* + 4, *x* is always the input (domain), and *y* is always the output (range).

Since this is a function, we can write the same equation using function notation: *f*(*x*) = 2*x* + 4. This means that the function of *x*, *f*(*x*), is equal to 2*x* + 4. 2 * *x* + 4.

If we evaluate *f*(*x*) = 2*x* + 4 for *f* = 7, *f*(7) = 2*x* + 4, we simply input 7 into the domain value, *x*, and run the function machine. What happened inside of the machine? As the 7 is input into the machine, the gears start grinding. It's first multiplied by 2, and that product is 14. When we add 4 to that, we arrive at the output of 18. Therefore, *f*(7) = 18.

Let's put in a different value. Let's evaluate *f*(-2) for 2*x* + 4. When I put the 2 into the machine, I multiply 2(-2) and get a product of -4. When I add 4 to that, I get 0. Is there any other number that I can input into my function that will produce a range value of zero?

Conclusion: *f*(-2) is the only domain value that will produce the range value of zero. This is the true mathematical definition of a function. So, a **function** has exactly one, and only one, output for each input.

The **domain** is the set of input values of an algebraic equation. The **range** is the set of output values of an algebraic equation. If we have an algebraic expression, *y* = -2*x* + 1, we can evaluate that **function** for different values of *x*. For example, we can input values into the domain of 1, 2, 3, 4, and then turn on the function machine. If we evaluate the function, we input the values into the expression:

*f*(1) = -2(1) + 1

*f*(1) = -2 + 1

*f*(1) = -1

Remember, every value of the domain has exactly one output. Every value of the domain has exactly one value in the range. Thinking in the other direction, every value of the range does not have exactly one value of the domain. But that's another story! Right now we're finished with algebraic functions.

Once you are finished with this lesson, you should be able to:

- Define the range and domain of a function
- Explain the properties of functions
- Recall how to use function notation

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

BackDid 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
1 in chapter 2 of the course:

Back To Course

Common Core Math Grade 8 - Functions: Standards5 chapters | 19 lessons

- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Computer Science 203: Defensive Security
- GRE Information Guide
- Computer Science 310: Current Trends in Computer Science & IT
- FTCE: Equations and Inequalities
- FTCE: Analyzing Data and Drawing Conclusions
- FTCE: Data Analysis & Visualization
- The Cybersecurity Threat Landscape
- Cybersecurity Policy, Governance & Management
- What is the ASCP Exam?
- ASCPI vs ASCP
- MEGA Exam Registration Information
- MEGA & MoGEA Prep Product Comparison
- PERT Prep Product Comparison
- MTLE Prep Product Comparison
- What is the MTLE Test?

- How to Determine the Number of Main Ideas in a Text
- Sequence of Events in a Narrative: Lesson for Kids
- The Square Root Property
- Number Theory: Divisibility & Division Algorithm
- Guided Reading Lesson Plan Template & Example
- Practical Application for Introduction to SQL: Views
- Computer Security Risk Assessment Computations: SLE, ALE & ARO
- Quiz & Worksheet - Slang Words in The Outsiders
- Quiz & Worksheet - Othello's Soliloquy
- Quiz & Worksheet - Adjugate Matrices
- Quiz & Worksheet - Double Angle Properties & Rules
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- 10th Grade Math Worksheets
- Math Worksheets for Elementary Students

- College Chemistry Textbook
- High School Physical Science: Homework Help Resource
- High School World History: Tutoring Solution
- High School Business for Teachers: Help & Review
- FSA - Grade 6 Math: Test Prep & Practice
- Three-Dimensional Geometry: Help and Review
- International Law in Politics: Tutoring Solution
- Quiz & Worksheet - Protein-producing Organelles
- Quiz & Worksheet - The Annexation of Sudetenland
- Quiz & Worksheet - Cognitive Processes in Learning
- Quiz & Worksheet - Divergent Thinking in Psychology
- Quiz & Worksheet - Educational Coaching Models

- Alfred Binet: Theory & Test
- Amusia: Definition & Symptoms
- Pennsylvania Science Standards
- What Is The Difference Between NGSS & CCSS?
- Ideas for Black History Month Projects
- What To Do If Your School Doesn't Accept Study.com Credit
- Community Lesson Plan
- Kentucky Homeschool Laws
- Homeschooling in Alaska
- Science Projects for Kids
- What is the PE CSET Like?
- 8th Grade Writing Prompts

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

Browse by subject