The graph below is an example of a step function. As you examine the graph, determine why you think it might be called a step function.
Do you see what looks like a set of steps? This is one reason why it is called a step function. It is better known as a discontinuous function.
Why do you think it is called a discontinuous function? Yes, it is not a continuous line, it stops and starts repeatedly.
So, the question may be, is it a function? Does it pass the vertical line test? Let's see!
It looks like the vertical lines may touch two points on the graph at the same time. However, take a look at the points. One is a closed circle and one is an open circle.
If you review our inequalities lesson, you will remember that a closed circle means that the point includes that particular point. But... an open circle does NOT include that point.
So, in this case, where it looks like the vertical line is touching two points, it is really only touching one point, because the open circle does not include that point. So, to answer our question, yes this is considered a function. It's not linear, and it's not quadratic. We call it a step function or a discontinuous function.
Let's take a look at our postage graph again.
This graph describes how much it will cost to send a letter depending on the weight of the letter. I've labeled the steps so that you better understand the explanation below.
Step 1: If the weight of the letter is over 0 oz and up to 1 oz (including 1 oz, since the circle is closed), it will cost 39 cents.
Step 2: If the weight of the letter is more than 1 oz (not 1 oz exactly because the circle is open) and up to 2 oz (including 2 oz since the circle is closed), then the price is 41 cents.
Step 3: If the weight of the letter is more than 2 oz (not 2 oz exactly because the circle is open) and up to 3 oz (including 3 oz since the circle is closed), then the price is 43 cents.
Steps 4-6 follow the same pattern as steps 1-3 described above.
As you can see, this graph tells you exactly how much your letter will cost depending on the weight. A discontinuous graph must be used because the price stays the same between ounces, but then changes to the next price as you reach a whole ounce.
Let's take a look at a few other discontinuous graphs and determine whether or not they are functions. These graphs may not look like "steps", but they are considered discontinuous.
This graph is a function because it passes the vertical line test. Each vertical line only touches the graph at one point. (Although it looks like it touches at two points at x = -3, since one circle is "open" we do not include that as a point.) Therefore, it is considered a discontinuous function. It is discontinuous at x = -3.
Let's practice creating and interpreting a graph for a discontinuous function.
A wholesale t-shirt manufacturer charges the following prices for t-shirt orders:
If I ordered 40 shirts and must pay $10 in shipping fees, then my total order will cost $610. (40 * $15) +10 = 610.
This would be a great problem for you to try on your own:
In many states a "sales tax" is added to most goods that you buy. The tax rate varies from state to state. Let's suppose that your particular state issues a sales tax on any goods purchased.
You are selling candy bars. The taxable amounts and tax imposed up to $1 are shown below.
Complete the graph to show the tax that imposed on candy bars.
Use the graph to answer the following questions:
1. A candy bar costs $0.55. What is the total cost with tax?
2. Your aunt purchased three candy bars at $0.55 a piece. What is the total cost with tax?
3. Someone purchased 4 candy bars at $0.55 a piece. They gave you $2 and a quarter. Is this enough money to cover the candy bars and the tax? Explain your answer.
The following is the completed graph for the discontinuous funtion.
The red circles indicate open circles.
1. If a candy bar costs $0.55, then the total cost with tax is $0.58. (0.55 +0.03)
2. The total cost of three candy bars is $1.74.
3. If someone gave me $2.25 for 4 candy bars, they would not have given me enough money. The total cost would be $2.31
Great Job! You have successfully completed the Step functions lesson.
You are ready to move onto the Exponents and Monomials Unit
Sign Up for Algebra Class E-courses
"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
Copyright © 2009-2013 Karin Hutchinson ALL RIGHTS RESERVED