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College Algebra: Operations on Functions


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:43
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 61 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: Composite Functions (5 lessons, $7.92)

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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So now we have a sense of what a function is, what they look like, how to graph them, how to manipulate them, even the notation for how to represent them. What I want to start thinking about now is really what's the algebra of functions; how do you combine them? And there are a couple of ways that are just really, really simple. For example, you can add them, you can subtract them, you can multiply them, and you can divide them. I want to just talk about those really fast because I think they're sort of straightforward ideas. Suppose I have two functions--one is f(x) = 2x + 3, and maybe have a different function, let's say g(x) = 4x + 8. Well, I can actually use these two functions to build a whole bunch of other functions, and well, what would those look like? Well, for example, I could add them, so we could look at the sum of these two functions, which some people write this way: f + g(x). That's just notation, by the way. Don't let that bother. That just means the function f + g(x). Still depends upon x, but I'm calling it f + g. If you don't like that notation, then forget it. Anyway, what would it equal? Well, you would just add these functions. We already talked about how to add algebraic expressions, you add like terms. I have 2x and I put in 4x that would be then 6x, and 8 and 3, that's +11. So there's a new function I made just by taking these two functions and combining them, in this case, combining with addition. You can imagine a similar kind of thing with subtraction where you subtract the functions. Multiplying them? Sometimes people write it this way: f x g(x), and it just means the product of those two things, and you can then actually foil that out if you wanted to. I don't know why you'd want to, but if you want to, this would be 16 and 12 would be what? That would be 28. And the last times the last is 24.
So here's a new function I built, and I built this new function by taking these two other functions and combining them in a way using multiplication. Again, not a big deal. You can imagine dividing them--you'll never believe the symbol for that. It's ()(x), and that would just be the quotient, . Now, there is a little word of caution I'd like to give you with respect to dividing two functions. If you divide two functions there's a little danger of actually having a change in the domains, because remember, functions are only defined when you can think of the values of x that can be plugged in to produce numbers. Well, if you have these two functions, for example, notice that the domain of each of these functions are all the reals, because I can take any number, multiply by 2, add 3. I can take any number, multiply by 4, add 8, not a problem. But once I divide them, once I have a denominator, there's a potential for the denominator to be zero, so in fact, the domain of this function, the quotients of these two things, actually is a little different, because whereas the bottom equals zero, it actually equals zero and x = -2. So, in fact, the domain will change sometimes when you divide, and in this case the domain of the quotient will be all the real numbers except x = -2, because x = -2 can't be plugged in here, not allowed, so I'd have a zero as a denominator.
Okay, so these are some common ways of combining things, just adding and subtracting, multiplying and dividing functions, just like you do with regular numbers. But it turns out there's a neat thing you can do with functions that you just can't do with regular numbers. I want to tell you about that, but before I do that, what I want to do is try to inspire this. So let's take a look at two examples. Before I can do that, let me tell you something else. Let me just do one more thing. Let's take a look at these two functions. How about x^2 - 1 for f and then g will be (x + 2)^2. What happens, for example, if I were to divide these things? Let's say I was looking at ()(x). What would that look like? Well, if I were to take the quotient I have to put the g on the top now, and then I divide that by the f. Now, that's a parabola, so that's defined everywhere, and this is also a parabola. That's defined everywhere. In fact, this is just a regular parabola that's shifted two units to the left. So when I take their quotients, all of a sudden this is not defined everywhere. The domain for this one--the domain here, is going to be all the real numbers except for when the bottom is zero. If you just say x = 1, then actually you have to be careful, because there's another root. It's + or -1. So you have to avoid both of those things.
So you see, when you take a quotient you really have to be careful with the domain. It's really important, whereas if I were to add or subtract or even multiply these things, the domain is probably not in harm's way. So there's another example to illustrate that.
Up next what I want to do is show you a brand new way of combining functions, a way that just doesn't exist with numbers. See, with numbers we can multiply, divide, add or subtract, just like we saw here with functions. But there's one thing we can do with functions that we can't do with numbers. It's a whole new way of combining functions together. Which there's no analog for numbers. I want to tell you about that and look at a whole bunch of examples coming up next. I'll see you there.
Relations and Functions
Composite Functions
Using Operations on Functions Page [1 of 2]

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