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College Algebra: Piecewise-Defined Functions

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About this Lesson

  • Type: Video Tutorial
  • Length: 9:46
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 105 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
College Algebra: Relations and Functions (57 lessons, $74.25)
Trigonometry: Algebra Prerequisites (60 lessons, $69.30)
College Algebra Review (30 lessons, $59.40)
College Algebra: Working with Functions (3 lessons, $4.95)

In this lesson, we will learn to define, recognize, write, graph and evaluate piecewise math functions. Piecewise-defined functions are unique in that they are defined to be equal to different things for different ranges of variable values. Thus, more than one expression defines the function (e.g. for x<2, f(x)=x^2 but for x>=2, f(x)=10x).

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/collegealgebra. The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

Thinkwell
Thinkwell
2174 lessons
Joined:
11/13/2008

Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...

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Recent Reviews

Nopic_dkb
EXAM 1
01/24/2012
~ Alexis7

exam 1

Nopic_blu
Awesome
11/04/2010
~ 245

If every math instructor were like you, people would enjoy math.

Nopic_grn
EXCELLENT
04/24/2010
~ joel8

THANK YOU!!!

Nopic_blu
Helped a lot
04/21/2009
~ thomas1

University instructor breezed over this in class. Over half of the class didn't understand the material yet he continued on. This lesson was worth both the time and money.

Nopic_tan
College Algebra: Piecewise-Defined Functions
02/16/2009
~ sharon

Awesome. Really helped me understand what was happening with piecewise-defined functions.

Nopic_dkb
EXAM 1
01/24/2012
~ Alexis7

exam 1

Nopic_blu
Awesome
11/04/2010
~ 245

If every math instructor were like you, people would enjoy math.

Nopic_grn
EXCELLENT
04/24/2010
~ joel8

THANK YOU!!!

Nopic_blu
Helped a lot
04/21/2009
~ thomas1

University instructor breezed over this in class. Over half of the class didn't understand the material yet he continued on. This lesson was worth both the time and money.

Nopic_tan
College Algebra: Piecewise-Defined Functions
02/16/2009
~ sharon

Awesome. Really helped me understand what was happening with piecewise-defined functions.

Okay, so if I just give you a simple equation we can talk about, well, what does the graph look like by plotting points and seeing roughly what looks like and sort of getting a visual picture for it. And usually in the examples we’ve seen they’ve been pretty smooth and nice. The absolute value one has that sharp v, but still, okay, whatever. What about some functions that have really unbelievable graphs? Let me show you one.
Here’s a function that just has an unbelievable graph. Now, I don't know if you can see all this, but I want to show it to you. First of all, the graph sort of comes down like this, you see. It keeps going, it comes down, and it gets to -2, 2. So you go -2, 2, and there’s that dot right there. But then it just stops, and it takes sort of this quantum leap; it’s like hyperspace, and it jumps right down to here, -2, -2, and it just keeps going. Now it’s constant for a while, and it’s constant all the way to 3, -2, and then it takes a quantum jump again up here to 3, 1.5 and then it starts to go down. So it has these jumps in it. It’s really, really weird. At first it’s falling very nice, looks like if I just covered everything else up you’re thinking, “Oh, that’s a nice, little, pretty function. Look at that. It looks so pretty and nice.” It goes through my hand; it comes out. But no, no, no. There is chaos under my hand, because in fact, what happens is it comes right to here, and then here it takes a quantum jump down, and then it keeps going constant for a while, and then it takes a quantum jump up. This thing is called a piecewise defined function, because it’s actually defined to be different things on different regions. Here it looks like some sort of nice curve, here it looks like it’s pretty much constant, and here it looks like some other nice curve, and I sort of piece them all together to make this really weird one. So a great question, and even if you’re not math fan you may be wondering--gee, can you write down an equation for that? What would it look like? Would it be really complicated? Well actually, the way you’d write it down--one way of writing it down anyway, is to write it down just like you see the graph, namely give a piece of it here, and then give a piece of it, and then give a piece of it. I’ll tell you that the function is actually three different functions. It’s one function on this domain, on this region, it’s a different function on this region, and it ‘s a third function on that region.
Let me try to illustrate this with a specific live example. Here’s what a piecewise-type function looks like. We’d have something like f(x) =… Now usually I write just a whole bunch of stuff with x’s right here, but not today. Today I’m going to write a big parenthesis bracket thing, and I’m going to write 3x2 + 1 and that’s for the case when x is bigger than or equal to 2. And then I’m going to write something completely different. -½ x + 14, and that’s if x is less than 2. Now this is a function that’s sort of an actual algebraic expression for a piecewise graph. And let me explain to you how you read this thing. It looks so complicated. The way it reads is the following. When you put an x in here you’re going to plug in to one of these two things. But to determine which one you plug into you have to see if x is bigger than or equal to 2, or if it actually is less than 2. So if I put in, for example, 5 into here, if I say what’s f(5), which one of these things do you go to? Well, you ask is 5 less than 2 or is 5 bigger than or equal to 2? Well, 5 is bigger than or equal to 2, so then I’m on this part, so I plug the 5 in right here, get 25 times 3 which is 75, add 1, I get 76.
If I say what’s f(2)? You ask, am I in this category or this category? Well, I’d be in this category because 2 is greater than or equal to 2, so I’d put 2 in here, and if I put a 2 in here I’d see 22 which is 4 times 3 is 12, and 1 is 13, and so on. What if I put in zero? If I put in zero in here, I’m not in this category, now I’m in this category, because zero is less than 2, and so I plug zero into this thing and I’d see 0 x -½ + 14 and I see positive 14. So the value of the function depends upon what part I am in terms of my domain.
So let’s see what that looks like graphically. The way I’ll do this is I’ll just graph each of these things separately. So if I graph this, we can do this by plotting points, for example, there are a lot of ways of graphing that, or you can remember that, in fact, x2 just looks like a parabola, and all I want to do is take the parabola but add one to it, so even though I’ll talk about this later, just for brevity, allow me just to draw this picture here. So that’s the graph of 3x2 + 1. And I could have found that by plotting some points, making a little table and plotting points, or knowing how the parabola looks, and then I just move it up by one unit, but we’ll talk about that later.
Now, let me take a look at the second person there. This just has x1, so in fact, it’s just a straight line, and this negative in front of the x tells me the line is actually going to be sloping downwards, and again, we’ll talk about all this later, don’t worry about it, but you can plot points to see, for example, if x is 0 you have to go all the way up to 14 and then you come down somehow, so if we had a straight edge we could do this very nicely. What would this look like? This is just rough, of course, but it’s going to be a line that looks sort of like this. So there’s the graph of the second thing. Now, I have to put them together. I just can’t put them on top of each other, because the truth is, I only want to look at this when x is bigger than or equal to 2. So what I do here is I just cut the part that’s bigger than or equal to 2, and I cut that part right here. And all I want is that part, the part that’s bigger than or equal to 2, and all I want is that part, the part that’s bigger than or equal to 2. You see? This part I don’t care about. And here, I’m going to take the line when x is less than 2, so I cut this part. You see I’m going to piece together these two graphs. And when you do that, I take this part and that part, when I put them together, I’m going to get a picture that represents the graph. So this graph is a very freaky, sort of disturbing looking thing, right? It’s this line that comes down, and then all of a sudden, the moment after you get to 2, you sort of jump up onto this parabola thing and this parabola thing sort of comes up. So remember, it’s not actually a perfect v, but this is a straight part here, and this is actually a curvy part of a parabola. But you can see the idea to do it. I graph the parabola and cut that region that I wanted, and I put it together with a part of the line.
Now, you can actually graph the whole thing together without doing this sort of gymnastics, and if you did that what you would get is something that looks like this. You would get a line for a while--here’s 2--and where would that line stop? Well, when x = 2 this would stop at 13, but then notice the action picks up exactly at 13. Remember, when we plug 2 into here we saw 13, and then I get part of a parabola, so then it becomes curvy. So that’s the graph of this, and you can see it’s sort of a weird thing with this bend where, in fact, something dramatically different happens if you’re to the left of 2 or the right of 2. This is what we consider a piecewise function. And if I want you to evaluate it, for example, at -1, we already plug in -1. You have to ask which category am I in. Negative 1, I’m in this region, so I go to the line, and so I’d see -1, I plug that into here, -1 times -½ would be just ½ + 14 is 14½. So at -1 I’d go up to 14½.
What about if I plug in f(2)? We did that earlier. Two is now in this category, x is bigger than or equal to 2, so I actually look at this part of the curve now, and I plug in 2 into there and I’d see 13. If I plug in f(4)--if I plug in 4, where do I go? Well, I’m in this part, so I’d see 42 which is 16, I’d take 16 times 3 and I’d add 1, so I’d be in this part of the wing. If I put in something like -5 or -4, -4 satisfies this, so now I’d plug it inhere. So in this case I would see 3(42 + 1), whereas here I’d see -½(-4 + 14). So you can see when you plug in a value for a function that’s defined piecewise, you have to make sure that you’re looking at the right piece. Okay, well, you can try to graph these things or just plot points and sketch in the values by making sure you’re on the appropriate part of that piecewise defined function.

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