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College Algebra: Domain and Range: An Example


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:16
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 57 MB
  • Posted: 06/26/2009

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: Domain and Range (3 lessons, $6.93)

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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You know, it's one thing to look at a graph and say, "Okay, what's its domain and what's its range," and just look for all the values for x where, in fact, the function sort of is colored in somewhere up or down, and the range, just ask for, you know, all the heights for which the function will finally cross at some point. But it's another proposition completely to look at just the algebraic expression for a function, for a graph, and ask just from looking at the algebraic expression, how do you go about determining its domain and range. Well, the domain actually is really important, and so we're going to look at a lot of examples of those in a second, but I thought we'd just take a moment to look at one function and find both its domain and its range by just looking at it algebraically.
So let's consider the following function: f(x) = 2x + 5. This is what's thought of as a linear function because I just have x appearing to the first power. And let's, first of all, find the domain and then the range. Okay, well, what's the domain? Well, if we're thinking about the domain, remember the thinking here is what are the permissible x values that I'm allowed to insert for x that would actually produce numbers? So what I have to ask is, are there any values for x, which would make this thing undefined? Well, if you think about it, no, there are no values for x which would make this undefined, because any number you can ever possibly think of can be multiplied by 2, and that quantity can be added to 5. So, in fact, this is defined for all real numbers, so the domain would be all reals. Sometimes, by the way, real numbers are denoted by-- it looks like a bold face R, but it's not, it's sort of two bars and then an R. So, all of the real numbers, so all the x values possible.
By the way, where would you be in a position where, in fact, that wouldn't be the case? Well, if you ever had a fraction as a function, then wherever the denominator were to equal zero, those would be points that you'd have to remove from the domain, because you can't have zero in the denominator. The other classic place to look is if you have square roots, you want to make sure the stuff inside the square root is always positive or zero. So any values for x which would make the inside of that square root negative, those would be points that you'd have to exclude from the domain. But here, very nice, clean, simple, no problem.
What about the range? Well, that's the target. What possible values can this take on as x goes through its domain? Well, if you think about it, it could be any value at all, because suppose you said to me, "Gee, I want to find out if this function will ever hit the number 23." Suppose you want it to hit 23? Well, all you have to do is say, "Let's find the x so that that y value will equal 23." Well, now you can solve this. How would you solve this? I would say 2x equals, I guess, it would be 18, and so therefore x would equal 9. And notice if I plug in 9, f(9)--so here's a little check, f(9)--I'm using this notation. That means wherever I see an x I plug in 9. That's going to be 2(9) + 5, and notice that's 18 + 5, which is 23. So I hit the y value, 23. But, in fact, you can see that I can hit any value that I wanted to, because I would just take 2x + 5 and set it equal to that value, I can always solve that equation. It's a linear equation, so I bring this over and divide by 2. So, in fact, the range, the y values that I possibly hit, turn out to be all the real numbers as well. So here's an explicit example where without looking at the picture at all I was able to see what the domain and the range is.
Now, let me show you the picture, because I happen to know what the picture looks like. This is actually going to be a straight line. So let me show you. Let me put in the axes here. Really fast--I'm not going to spend too much time on this. But if here's the graph, here are the axes. Well, it turns out that this graph looks something like this, roughly speaking. So it's a line that shoots out this way, and you can see that every single value of x is allowable to plug into this line. Now, of course, it goes right through my head and stuff, but it keeps going, even though I'm standing there, and similarly, over here. So, in fact, you can see that the domain is everything. If you sort of take this thing and project it down, if you project it down, you sort of get everything. Similarly, if you now look at the y values you get, you can get really, really high y values, but you also get really, really low y values, so we get all the y values. So now you can see graphically that, in fact, the domain and the range, in fact, are both all the numbers, but we're able to do it without even looking at the graphs.
What I want to do next, is to take a look at a lot more complicated functions, and there the question is, find the domain. Find the allowable x values that we can put in this input. I'll see you there.
Relations and Functions
Function Domain and Range
Domain and Range: One Explicit Example Page [1 of 1]

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