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College Algebra: Inverse Functions

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About this Lesson

  • Type: Video Tutorial
  • Length: 8:43
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 94 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Algebra: Exponential and Logarithmic Functions (36 lessons, $49.50)
College Algebra: Function Inverses (4 lessons, $5.94)

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 athttp://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.

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So, we've spent a good chunk of time thinking about, together, the notion of a function. That's where you sort of have an input, you put something in and then something else is spit out. But now I want to think about sort of the opposite question. Suppose that you put something in, the thing is spit out and now you want to undo that process. Suppose you're given the spit and you want to go backwards and find out what was the input that sort of spit that out. So how do you reverse that process? How do you go backwards? Think about like a coding kind of thing. You take a message and somehow you code it, using x² and you get something out and now you want to decode it. You want to take the output and do the process in reverse and then all of a sudden you'd have what the original x was.
Well, that is the idea and the philosophy behind finding inverse functions. So an inverse function, basically, is a function that undoes another function. So now how do these things work and what do they look like? Well, basically, inverse functions sort of behave like this. Suppose I have a function f(x) and I have some point x and if I see where f(x) sends it, it sends it to something--it's like a little machine and it sends it to f(x). So I have now another number. So I put in like 5, it spits out something else. Now what I'd like is to build a new function, let me call it like g(x) which has the property that if I use this as the input and I map it with g, so now I'm going to use a g. What would it look like? It would look like g of this thing, that's the input now. See I'm inputting the output of the other machine. That should give me the original x back.
So this maybe looks sort of confusing notation, but think about it. I put in the input x, it goes off and gives me f(x) as output. Now I take that output and put it inside a new functions and it should spit out the original x. If you notice that's a composition of function, that's gof, right. It's g composed with f. So we're back to compositions of functions, and I want that to be x. Similarly I want f to be the untangling of g. So if I did this thing starting with g, if I started with a point x up here and first did g to it, so I first did the decoding process and then I hit it with f. So, then I put in f of and I put in the output, I'd still want to get back to where I started. This would still be x. Again I see now fog, f composed with g of x equal x.
Well, this all sort of looks kind of complicated and indeed it is complicated looking, but the important thing is I just want to find a function that will undo another functions. Let's try an example just to sort of get our feet wet and see just the basic idea of what I'm talking about. Because undoing a function sounds a little bit funky. So, let's suppose I have function f(x) and it equals 2x - 3. Okay. Now, what I want to do is I want to find function that untangles that. So, what would I do? Well, for example, let's just look at some example here.
So, what would f(1) equal? f(1), if I put in 1 would give me -1. f(3) would give me what? This would be 6 - 3 would be 3 and so forth. So, what I want is I want another function that has the property that if I input 3, it'll spit out the original 3. If I input -1, it'll spit out the original 1. For example, let's put some more points in. What if I put in like -2? If I put in -2 here, I see -4 and 3 is -7. So I want this new function, when I input 7, it outputs -2. Do you see how it would untangle? Well, let's try this function, g(x) = x + 3 all divided by 2. That's a different function. Let's see what happens if we input into here the output of here. Just for fun. If I put in a -1 here, I have a -1 + 3, that gives me a 2, I divide by 2 and look what I get. I get 1 and that's what I started off with. That's pretty cool.
Let's try this one. So, g(3)--our fantasy is that it should give us back the original 3. Let's see. If I put in 3 here, 3 + 3 is 6, 6 divided by 2--we get 3. What about this one, g(-7)? Is it possible that we actually get -2? If I put in -7 here plus 3 gives me a -4 divided by 2, is in fact -2. This is really cool. This function seems to decode that function, because if I take any output value and put it as input I actually get back the original thing. Really cool. Now, can we see that sort of as a general idea? The answer is--well, let's try it. Instead of putting in particular numbers, let's just actually plug the whole function in. So, let's do that composition business. Let's look at fog. This is composition now.
This means, f composed with g, f(g(x)). So, what does that mean? It means wherever I see an x in the f function, I'm going to replace it by g, which is this whole function here. Let's plug this in wherever I see an x. So this would equal--I see 2, so I write a 2 times x. But now in place of x I write all that stuff. So, I write x + 3 all over 2. So there's the x and then -3. All I'm doing is I'm evaluating f at g. That means wherever I see an x, I'm going to insert all of g right in there and I get this. Well, notice the 2's cancel. So, I just have x + 3 -3, x + 3 - 3 = x. So, I start with x and these two functions sort of cancel each other out and I end with again. Whatever I start with I get something. When I plug it into the other thing, I get back to what I started with. So, you can see these two functions untangle each other. These are called inverse functions.
Now, does every function have an inverse? The answer is, darn it, no. There are functions that don't have inverses. Let me give you an example. f(x) = x². The good, lovable, standard parabola. How could this thing not have an inverse? Well, let's think about it. Suppose I input 2 and that would, of course, spit out 4. Now what I need is some machine that will take the 4 and spit out 2. So, maybe you're thinking how about square root. That wouldn't be a bad idea. How about the square root function? Maybe we should think that g(x) equals the square root function. Because notice g(4) = 2. Looks good. But let's try this, f(-2). If I put in -2 and square it, I still get 4. But look what happens when I plug back the 4 again, I still get 2.
So, this is not giving me back the exact value I got before. The problem is because there are different values of x's here that lead to the same y value. So, once I get the same y value, I can't go backwards. Because if I start with a 2, that leads me to 4, but if I start with a -2, that leads me to 4. But if I start with 4, how can I go backwards? I don't know where to go. Do I go to 2? Do I go to -2? Both roads get back to 4. So, which one did you have in your mind? I can't tell. So this function can't be undone. Because once you square something, you don't know exactly where you came from. You could have come from a positive number or you could have come from the negative version of it. So some functions actually can't be untangled and we'll take a look at why and how you can tell which functions can be untangled and which functions can't be untangled. This whole notion is the notion of finding the inverse function.
Exponential and Logarithmic Functions
Function Inverses
Understanding Inverse Functions Page [2 of 2]

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