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College Algebra: Graphing the Inverse


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About this Lesson

  • Type: Video Tutorial
  • Length: 6:12
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 66 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 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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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 or visit Thinkwell's Video Lesson Store at

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


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So suppose someone actually gives us the graph of a function? How can we determine, first of all, if it has an inverse, and then how can we find out what that inverse is graphically? Let's forget about the algebra stuff for a second. Let's just think about visually what's going on. Well, the first thing we have to look at--so here's a function--the first thing we have to ask ourselves is does it have an inverse? Well, that's asking the same thing as saying is it a one-to-one function? Which is the exact same thing as saying does it pass the horizontal line test? We know it's a function because it passes the vertical line test. By the way, the function is this here, this dotted line is just the line y = x, which I'll use in a second.
Anyway, does it pass the horizontal line test? It sure does, because notice at every single horizontal line, the curve only hits the line at one point at most. So, in fact, this does have an inverse, it is one-to-one. How do I find it? Well, all I have to do is reflect this picture over the y = x-axis, the y = x line. Because I want to switch the roles of x and y. I want to untangle. So you can think about that as just taking a mirror and putting it right there and reflecting, and then you can see how the curve should go. It should go on the other side there. Over here in this part you can see how it's supposed to go. It's supposed to go just like that.
So if you graph that, you just have to graph the reflection of what you see, so as best you can you draw the reflection--and I'll do my best --Hey, that's not bad. Look at that. You see how I just took this picture and reflected it? And so this function were to be called f(x), then this purple function would be notationally called f inverse of x, and all I did was flip. Not a big deal.
Let's try one more together. This looks sort of like a cubic function to me. It has that cubic feel, doesn't it? Sort of up and over and up. Three little bends. Sounds cubic, but who knows. It might be something that's sort of a cubic in disguise. First of all, it doesn't have an inverse function. Does it have an inverse function? Well, we can see that by seeing if it passes the horizontal line test. Does the curve only hit a horizontal line at at most one point? Yep, sure does. There's no backup onto itself like a parabola would have. So, in fact, this does have an inverse function, and to see what that is we can just take this, put it on here, and you can see that part of it's going to come down there and so on. You'd have to just look at the inverse so you'd draw this picture. I'll do my best here. Okay, so this would actually be f inverse of x. There you go. If this were f(x). So you can see that visually all you do is just flip the picture, and what's happening is, instead of x and y, you flip it, and you have y, x. It undoes each other.
Let's try one last example just on the fly here. I don't even have a visual mock up for it, but I'm going to make it for you right now live and in person. So this is the function f(x) = x². The first question is, does this have an inverse? So the question is does this pass the horizontal line test? Is this function one-to-one? And you can see the answers easily seem to be no, because in fact, for this particular y value, I don't know where it came from. Did it come from -2 or did it come from 2? Both points lead to 4.
Another way of seeing this is to actually try to do it. Suppose you didn't do that. Suppose you wanted to push forward. You said, "Darn it, I'm going to find the inverse. I don't care what you say, Ed Burger, I'm going to find the inverse, and here's how I'm going to do it. I'm going to put in the line y = x and I'm going to flip over it. That'll teach you." I don't know why you're so angry with me, but okay. There's the y = x. Let's take this picture and now flip it over that line. Now, it' s a little hard to visualize, but give it a try. I think you'd see this. Do you see how this wing would sort of flip over and form that wing? And this part of the wing here would flip over and be here? And that wing would flip over and be here? So you can see this would be the inverse.
Now, is that inverse really a function? Well, notice, no, it's not. It fails the vertical line test. The green fails the vertical line test. So, in fact, this does not have an inverse. And now you can see where the horizontal line test comes from. Do you see it? The horizontal line test is really just the vertical line test after you flip over this line. Watch. Flip over this line. Do you see that if you just take this pink line and flip it over the y = x line, what you see is actually the green line. This piece gets mapped to here, and this piece gets mapped to there. So actually, the horizontal line test is nothing more than the good, old-fashioned vertical line test for testing functions for the inverse. After you flip, horizontal becomes vertical. So just by doing this test right here, you can see that the inverse will not be a function. So, in fact, the square, as beautiful and simplistic as it is, is not invertible--it has no inverse. Sorry.
Anyway, have some fun with mirrors and inverses and see if you can graph the inverse function.
Exponential and Logarithmic Functions
Function Inverses
Graphing the Inverse Page [1 of 2]

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