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College Algebra: Domains of Radical Expressions


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About this Lesson

  • Type: Video Tutorial
  • Length: 6:47
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 72 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Equations & Inequalities (50 lessons, $65.34)
College Algebra: Inequalities: Rationals, Radicals (3 lessons, $4.95)

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 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

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Sometimes when someone hands you some sort of algebraic equation or a function or something with symbols in it, you want to know when is that thing defined. When does it make sense? For example, if it were a fraction and if there were x's in the bottom, any value of x that would make the bottom equal to 0, those are values where you know the whole thing is not defined. Now, with square roots, you know that you can't take the square root of a negative number. So, any time you can get a negative under a square root, you know the thing is not defined.
So, here's an example to sort of illustrate this sort of circle of questions. Suppose I tell you that y is equal to the . The question is, I want to know for which values of x is this whole thing defined. Now, you may remember this is called the domain. So, the domain is just the collection of x's, which make this thing defined. Which means, when I say, make this thing defined, what I mean is the collections of x's that I can plug in here for which this thing would really be a number. For example, is the value 0 in the domain? Well, let's think about it. Am I allowed to have x = 0? Well, if x were 0, look what would happen. That would go away and I'd be taking the . Ouch. You can't do that. It's not a real number. So, 0 is definitely not going to be in the domain or in the domain of this function.
So, how would I find the domain of this function? Well, since all I see is the square root, I know the only principle here at stake is that the thing under the square root, that thing has to be positive. So, you see that if I ask you to find the domain of this function, it actually converts to an inequality. The inequality, which says, okay, the domain is all the values for which this inequality holds. Namely, x² - 16 is bigger than or equal to 0. You see? Otherwise, if this is negative, I can't take the square root. So, even though the question asked about find the domain of this expression, really it converts to a question of solving an inequality.
Well, now we're back home where we know what we're doing here a little bit. I want to try to factor. It's a quadratic. So, I have x and it's a difference of two perfect squares. So, it's great. It's (x + 4) (x - 4). What do I do now? I'm going to find out where this thing equals 0 and use that as sort of the end points of a sign chart. So, I'm going to make sign chart right now. So, let's see. Where does this thing vanish? What values of x make this equal to 0? Well, since I have a product of two things that combine to give 0, either this is 0 or that is 0. If this is 0, x + 4= 0, that means x has to be -4. So, I have -4. I'll mark that here. If this term is 0, x - 4, that means that x has be 4. So, there are two values, -4 and 4, and I'll write 0 here to indicate that this thing is 0 at that point.
In fact, if you want, you might want to put up those firewalls. Firewalls just to really drive home the fact that this breaks up these 0's or if they were denominator, where the denominator equals 0, where it's undefined, break up the real line into, in this case, into three pieces. Whatever the sign of this thing is at any point in here, it will remain that same sign throughout this whole region, similarly here and similarly here. Why, again? Well, suppose that this point, this thing were to be negative. Then right over here, at this point, it was positive. Well, how do you go from a negative to a positive? Somewhere there has to be a value in there, which makes this 0. So, we'd have another 0. But wait, we found all the 0's. They're here. So, this is all the same sign. This will be all the same sign. This will be all the same sign.
So, how do we find it? Just pick points. For example, let's pick a point out here, like 10. Let's plug in 10 and see what the sign of this is. I don't actually care about the actual number. I just picked 10 out of the hat. So, don't worry about computing the number. If I put a 10 in here, 10 + 4 = 14, but all I care about is its sign. It's positive. If I put a 10 in here, 10 - 4 that's whatever, but it's positive. So, it's positive times positive, that's positive. So, this whole region is a positive region. This is where all the people that see the world with rosy glasses live. Now, what about here? Between -4 and 4 I've got to pick a point. How about just 0? That's in there. It's pretty easy to see. If I put in a 0 for x, this is just 4. If I put in a 0 for x here, that's -4. Positive times a negative is a negative. So, in fact, in this region this is where the pessimists live, in the negative region.
What about here, if I pick -20? -20 + 4 is still negative and -20 - 4 is also negative. It'll be -24. So, a negative times a negative is a positive. So, the net result is a positive region. So, there is my sign chart. Great. What do I want? I want to find out where these things are greater than or equal to 0. So, that's off of these wings here. So, the solution, if I were to label it, would be all these places, the positive region. Now, what about the end points? Do I want to include them or not? Well, those are places, which make the thing 0. Am I allowed to equal 0? Yes, because I can have a square root with a 0 inside, that's okay. So, I'm allowed to equal 0. So, I'll include it here and I'll include it here.
So, this is a visual answer. This is a graphical answer of the question, what's the domain, any point in here. But notice if I pick any point in here and plug it back in, that will make a negative under the square root, which is not allowed. So, how would you write that? You could write it this way. I could say, well, it's all the points from negative infinity to -4, including -4, but not including infinity. In addition, all the point from 4 out to infinity and we take either one or the other. Any of them will work. So, it's the union of those and so I use the little happy face union sign.
So, the question was "What's the domain of this function?" Where are all the allowable values of x? It turns out any x that lives in this region or in particular, graphically, any x that lives out in this wing or that wing. So, that tells you how to find domains when you've got square roots. What you've got to do is actually set up an inequality, solve away. See you soon.
Equations and Inequalities
Inequalities - Rationals and Radicals
Determining the Domains of Expressions with Radicals Page [1 of 2]

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