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College Algebra: Finding the Inverse of a Function

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About this Lesson

  • Type: Video Tutorial
  • Length: 3:40
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 39 MB
  • Posted: 11/19/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra Review (30 lessons, $59.40)
Algebra: Exponential and Logarithmic Functions (36 lessons, $49.50)
College Algebra: Finding Function Inverses (2 lessons, $1.98)

This lesson will teach you how to find the inverse of a function [f-1(x)] when you are given the function [f(x)] as a formula algebraically. Some functions, however, have no mathematically defined inverse. Professor Burger will show you how to recognize when a provided function has no inverse. For example, a parabola function cannot be inverted.

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

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 http://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.

About this Author

Thinkwell
Thinkwell
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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 www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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

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Recent Reviews

Nachan_homepage
Very clear and informative
01/17/2009
~ nachan

Professor Burger does a great job explaining how to find the inverse of a function.
At first I thought it would be very difficult to solve the inverse of a function but it turns out it is as easy as solving for y and x. He teaches how to find the inverse of functions with fractions and with negatives, he gives many different sample problems so you understand every possible way to solve an inverse of a function.

Nachan_homepage
Very clear and informative
01/17/2009
~ nachan

Professor Burger does a great job explaining how to find the inverse of a function.
At first I thought it would be very difficult to solve the inverse of a function but it turns out it is as easy as solving for y and x. He teaches how to find the inverse of functions with fractions and with negatives, he gives many different sample problems so you understand every possible way to solve an inverse of a function.

How do you find an inverse, if it actually exists, if you’re given the function algebraically? We know how to do it if we have a visual picture. We just sort of reflect over the y = x line. But how do you actually perform this task if you’re just given the function algebraically? Well, it’s actually sort of fun--I think it’s fun. I’m going to use the y notation instead of f(x) for a second. But you can think of this as f(x) = 4x - 3, but remember, it’s the same thing as y = 4x - 3. This is the old notation, in fact. It’s real easy. All you do if you want to untangle this, is just flip the roles of x and y. Remember, it’s a reflection over the y = x line. So all I’m going to do is literally flip x and y roles and then solve back for y. If I can solve back for y and I get a function, then that function is the inverse. If I can’t solve it for y, then it has no inverse.
So, for example, what I do here is I flip the roles of x and y. So notice, this is not going to be the same equation. I’m going to change the equation, but I just flip roles. So now I see x = 4y - 3. And now I just solve for y. If I solve for y I bring the 3 over, so I have 4y = x + 3, and I divide both sides by 4, and I see y = . Since I was able to solve it, I see this is the inverse. So if this were f(x), I now see this as f-1(x). Pretty cool. That’s all there is to it. Let’s try some more.
So suppose I’m given the function f(x), and I’ll write it as y = . What would the inverse be? I just flip the roles. I take very x, make a y. I take every y, make it an x. So I’d see x = , and now I’ll solve for y. How would you solve for y here? Well, there are a variety of things you could do. I guess you could cross-multiply, think of x as being . The other thing you can do is multiply everything through by y. I think I’ll do that. I’ll multiply every single thing through by y. So on this side I see yx and on the other side I just see -6. Now, I divide those sides by x, and I see y = . That’s a perfect fine function, so, in fact, that would be the inverse function of this function. So if f(x) = , it turns out the inverse is itself. You can check that. Compose the two functions, you see you just get x.
All right, let’s try another one. So f(x)--but again, I’m going to write that just temporarily as y--=x². So what do you do for the inverse? You just switch the roles. So x = y², and now you want to solve for y. Well, what do you have to do? Well, I take square roots, but I’ve got to take plus or minus square roots. Oh-oh. This is not a function because I have two values. If I put in one x value this wants to spit out two y values. Well, that’s not a function. So, in fact, this cannot be solved for one particular value given one x value, and so, in fact, this is not invertible, and of course, that’s just a good, old-fashioned parabola, and I already told you that that can’t be inverted. So, in fact, we’re just seeing another example of it.
Up next we’ll try even some more exotic functions, switching x and y, solving for y, seeing if we can do it. I’ll see you there.

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