The Inverse That Shouldn’t Exist?!

Added: 2026-05-11

[00:00:01]Hello everyone. In this video, we're going to be finding the inverse of a function. So f ofx is ln x / x and we're going to find the inverse of f.

[00:00:14]So to be able to solve this problem, we're going to be using a very very special function and we've done similar problems before. So if I can find them, I'll try to share a link down below.

[00:00:26]Okay, great. So let's go ahead and start with what is given, which is f ofx equals ln x /x. For some reason, every time I start recording a video, one of my cats start meowing. I don't know why.

[00:00:40]They shouldn't be hungry. They have water. They have food. They have everything. The only thing they don't have at this point is the attention. Of course, they're not getting that right now. Anyways, so to be able to find the inverse of a function in general, this is what we do. First we replace f ofx with y. So now we can write it as y = ln x /x. And then we're going to go ahead and switch x and y. Okay? And you can do this in two different ways. You can switch x and y and then solve for y. Or you can solve for x and then switch x and y. It's the same idea. So let's go ahead and actually hm how should we do this? Uh maybe we can do this uh directly. Solve for x first. Okay, let's do that. So I'm going to go ahead and write this on the left hand side. My goal is to solve for x.

[00:01:34]Okay, first step, solve for x. Okay, that's what I'm going to do first. To solve for x, again, we're going to be using a special type of function to be able to do that. Obviously, uh even though you wrote it this way, I don't want one overx.

[00:01:56]That's not a very good thing to have. Or is it? Yeah, actually I can deal with that. Let's find out. Okay, so let me go ahead and write the x 1 /x as x ^ -1 * ln x and that is equal to y. Now we want to get we want to put this in a different form which is actually going to be the following form t * e to the t and the reason behind that is this is called the product log I mean the function that I'm going to be using uh which is also called lambert's w function. So anytime you apply lambert's w function on or to t * e to the t you get t as a result. That's why this function is called the product log because if you remember with the log function or the ln function if if you apply it to e to the t we get t. So from an exponential we get the t the argument from a product like this we have to apply a special function we cannot express it like the ln function that's why we have to call that w okay so we need to bring this to a t * e to the t form how do we do that okay here's what we're going to do first we're going to bring an x ^1 here by using properties of logarithms. First step, we're going to multiply both sides by -1.

[00:03:28]Okay, that's going to give us negative y and we're going to take this -1 and bring it over here because properties of logs tell us you can do that. So this becomes x ^ -1 * ln x ^ -1 = - y. So far so good, right? Again, my goal is to solve for x, right? That's what I'm going to do first, and that's what we're trying to do. So, this is still not in the t e to the t form, but we're so close. How? Well, you can do a couple things here. For example, if you call this t, if ln x ^1= t, then from here x ^1 just becomes e to the power t by properties of logarithms or just by definition. And because ln is base e, so x ^1 can be replaced with e to the t. So you're going to get t e e to the t = y. That's actually one way to look at it. Or you can use an identity x ^1 can be written as e ^ ln x ^1 and when you do that you get something like this ln x ^1 * e ^ ln x ^1 and this just becomes your t. Okay, make sense? Okay, great. Now substitution is a little safer I would say. So let's go ahead and use that because this kind of looks a little complicated but that's the whole idea right bring it into this form t e to the t. Great. So now we got it and now we we can apply lambbert's w function. Let me go ahead and move this a little bit uh away so that we can apply lumbers easily. So let's put a big w here and big w here. Okay. and let's erase this and put a nice equal sign. Great. Now, when you apply Lambert's W function to T the T, you know what you're going to be getting, right? It's going to give you T. Okay, great. Now, the right hand side is W of - Y. But wait a minute, we said that we were going to solve for X, right? Well, we didn't, but we're close.

[00:05:55]Now, we have to back substitute. What is t? T must be replaced with ln x ^1. ln x ^ -1 = w of - y. And of course you can move this back. So make it ln x = w of - y. And then multiply both sides by -1 to get ln x because it is better than negative ln x. And we get that. Now we're so close. Remember e to the power ln x is equal to x. So let's go ahead and do e ^ both sides. e to the power ln x = x which is e ^ w of - y. You can totally forget about this and just focus on this. And remember our goal was to solve for x and we did. And guess what? By solving for x because remember we had y = f ofx or f ofx= y and by solving for x we actually found x= f inverse of y because f inverse basically does the opposite. It switches x and y. So this expression right here is equivalent to f inverse of y. Let's go ahead and write it that way. f inverse of y = e ^ w of - y. But you got to remember what was the original problem? It was asking for f inverse of x. Most of the time we like to express our functions the original one and the inverse or any other function that we use. We like to express them in terms of x. x is usually the independent variable and y is the dependent variable. That's why in this expression, we're going to go ahead and replace y with x. And remember what I told you originally, there are two ways to do it. You can uh switch x and y and then try to solve for y. Or you can solve for x and then switch x and y. That's what we're going to do now. This will give us f inverse of x equals e ^ w ofx.

[00:08:05]So whatever your function is f ofx this will give you the inverse. And of course we there are two branches of the lambs w function. I made quite a few videos about lambert's w function. Go ahead and check them out. Also make sure to check out a plus bi which is another channel of mine that focuses on complex numbers.

[00:08:26]And I also have a short channel. And this brings us to the end of this video.

[00:08:30]Thanks for watching. I hope you enjoyed it. Please let me know. Don't forget to comment, like, and subscribe. I'll see you next time with another video. Until then, be safe, take care, and bye-bye.