Home
#### Algebra and Pre-Algebra Lessons

Algebra 1 | Pre-Algebra | Practice Tests | Algebra Readiness Test
#### Algebra E-Course and Homework Information

Algebra E-course Info | Log In to Algebra E-course | Homework Calculator
#### Formulas and Cheat Sheets

Formulas | Algebra Cheat Sheets

Home » Functions »
Vertical Line Test

# Is the Relation a Function?

Using the Vertical Line Test

## Example 1

## Example 2

## Example 3

Click here to practice identifying functions using this special test.

# Like This Page?

## Other Pages You Might Like

Using the Vertical Line Test

A function is a relation (a set of ordered pairs) where the value of one variable depends on the value of the other variable. However, in a function, each input (x coordinate) may be paired with only ONE output (y coordinate).

Let's look at a couple of examples to clarify this definition.

Let's look at our relation, b that we used in our relations example in the previous lesson.. Is this relation a function? Is each input only paired with only one output?

There are actually two ways to determine if a relation is a function. One way is to analyze the ordered pairs, and the other way is to use the **vertical line test**. Let's analyze our ordered pairs first.

Since each input has a different output, this **can** be classified as a function.

Let's verify it with the **vertical line test**. This special test is used when you graph the ordered pairs. You imagine a vertical line being drawn through the graph. If the vertical line only touches the graph at one point, then it is a function. If the vertical line touches in more than one point, then it is NOT a function.

Let's graph our points and use the vertical line test to prove that this is a function. I drew the vertical lines (output) on the graph to demonstrate what it would look like. Unless asked, you really don't need to draw the vertical lines, you can just imagine the vertical line, or I have my students use the edge of a sheet of paper and move it across the graph.

Since each vertical line only touches the graph at one point, this relation can be classified as a **function**.

Let's take a look at another example.

Is the relation s, a function?

s:{(-3, 2) (-1, 6) (1,2)

Let's ask ourselves: Is each input paired with only one output? (This one is a little tricky!)

YES, each input is paired with only one output. I know that you see an output of 2 twice, but each is paired with a different input. The outputs can be the same, as long as the inputs are different. Let's look at the vertical line test.

Yes, this is a function - remember that the outputs can be duplicated as long as the inputs are different.

Ok... I hope that you aren't confused by that last example. If you are, take a look at this last example. You will see a relation that is **NOT** considered a function. Hopefully this will help build your understanding.

Is the relation, c, a function?

Let's ask ourselves: Is each input paired with only **one output**?

c:{(3,3) (-1,0) (3,-3)

NO, look at the ordered pairs - there are 2 ordered pairs with input of 3 and they have different ranges! Think about our little function machine - you cannot put in a 3 and get out a 0, and then put in another 3 and get out a -3. Each time you put in a 3, you should get the same answer.

Therefore, this relation is NOT a function. Let's look at the vertical line test.

So, how do you feel about identifying whether or not a relation is a function? I know it can be a little confusing.

Just remember: If you have duplicate inputs and they are paired with different outputs, then the relation is not a function. The vertical line test is always a good technique to use if you are unsure or want to justify your answer.

If you need more direct instruction, check out the *Algebra Class E-course *for video tutorials and more practice problems.

Sign Up for Algebra Class E-courses

Click here to retrieve a lost password.

Custom Search

Most Popular Pages

*"I'd like to start off by relaying my sincerest gratitude for your dedication in teaching algebra. Your methodology is by far the simplest to follow, primarily because you take the time to include the small steps in between that most other teachers leave out.*

*It helps to know why you are doing something. I am 45 and heading to college to get my BS in Business. I need to brush up, hence the visit to your site. Great Job!"*

Jimmy - United States

*"I stumbled onto your site after I found out that I needed to use some fundamental algebra for an assignment. Turns out I had forgotten some things and your great site helped me remember them like "that" (snap of fingers). The organization of the site let me find exactly what I was looking for so easily. Kudos to you for maintaining such a great resource for students of all ages!"*

Tom - United States

*"I just wanted to write and basically thank you for making such a wonderful website! I'm 20 years old and about to take a basic placement test for college. I wanted to brush up on my Algebra skills and I stumbled upon your site. I'm amazed at how simple you make it and how fast I'm remembering Algebra! I don't remember getting most of the answers right when I had an actual teacher in front of me teaching this. Thanks a lot!"*

Elizabeth - United States

*"I am a pensioner living in South Africa. I stumbled on your website, the best thing that could ever happen to me! Your course in Algebra has helped me a lot to better understand the different concepts. Thank you very much for sharing your skills for teaching math to even people like me. Please do not stop, as I am sure that your teachings have helped thousands of people like me all over the world."*

Noel - South Africa

*This is an amazing program. In one weekend I used it to teach my Grade 9 daughter most of the introductory topics in Linear Relations. I took her up to Rate of Change and now she can do her homework by herself.*

Reg - United States

- FREE Solving Equations E-course
- Algebra Class E-course
- Algebra Class Products
- Algebra Practice Test
- Algebra Readiness Test
- Homework Answer Calculator
- Practice Worksheets

- Site Map
- Pre-algebra Refresher
- Solving Equations
- Graphing Equations
- Writing Equations
- Systems of Equations
- Inequalities
- Functions
- Exponents & Monomials
- Polynomials
- Quadratic Equations
- Algebra 1 Final Exam

- SAT Online Course
- Algebra Cheat Sheets - Very Popular!!
- Algebra Formulas
- Online Resources
- Contact Me
- I Want to Hear From You!
- Algebra Blog
- About Me

? ## Subscribe To This Site

Then why not use the button below, to add us to your favorite bookmarking service?

Copyright © 2009-2015 Karin Hutchinson ALL RIGHTS RESERVED

## Comments

We would love to hear what you have to say about this page!