College Algebra: Finding the Inverse of a Function

Very clear and informative

01/16/2009

~ 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.

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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.