This Function Has a Secret…

Hello everyone. In this video we're going to be solving a functional equation. We have 2 times f of x plus f of negative x plus f of 1 over x equals x. And we're going to be solving for f. In other words, we're going to find an expression for f of x in terms of x. So let's see how we can solve a problem like this. So in this expression, if you noticed, x and negative x are inverses. x and 1 over x are reciprocals. So that's what we're going to take advantage of. For example, we can go ahead and replace x with negative x on both sides. So let's start by replacing x with negative x. That will give us 2 f of negative x plus negative negative x will be positive x. And here we're going to get f of negative 1 over x. And of course on the right hand side we should be getting negative x. So now notice that a new term came up like negative 1 over x. How do we get rid of that? One of the things you can do is you can try replacing x with 1 over x. That will give you 2 times f of 1 over x plus f of negative 1 over x. If you replace x with 1 over x. If you replace x with 1 over x, 1 over 1 over x will be f of x. And on the right hand side we have 1 over x. Awesome. So now notice we have f of x, f of negative x, f of 1 over x and f of negative 1 over x. So it looks like we have 4 variables but 3 equations. So we need a 4th equation which we can come up with if we replace x with negative 1 over x. The only thing we haven't done so far, right? So if you do that, we'll get 2 f of negative 1 over x plus f of negative x will give us f of 1 over x. Plus f of negative x equals negative 1 over x. So now notice that we have 4 equations and 4 unknowns. We should be able to solve for this, right? So why don't we just go ahead and name these something. Because instead of just repeating the f of f of this f of that. We can just call them a, b, c, d which will be a lot easier to solve, don't you think? So let's go ahead and call this a. Call this b. Call this c. The only thing that's different is this one. Let's call that d. So we get the following. 2a plus b plus c is equal to x. That's our first equation. The second equation is, if you notice that this is going to be a b. So we're going to get 2b or not 2b, that's the problem. Plus f of x which is a. Plus f of negative 1 over x which is the new thing d. And that's equal to negative x. And then we have 2 times f of 1 over x which is 2c. And then we get f of negative 1 over x which is d. And then the next thing is going to be f of x which is a. And this should be 1 over x. And the final equation is going to give us 2 times f of negative 1 over x. Again, that is d. And then b, a, d, c. So hopefully you can tell which is missing. And negative 1 over x. Great. Now, how do we solve for a? That's a good question, right? Maybe if we add everything up, do you think that's going to help? We can give it a try. Looks like we have some type of symmetry here. If we add everything up, we're going to be getting 2a plus a plus a which is 4a. And then we have 4b. And then we have 4c, right? And then 4d because everything will appear, you know, 4 times. And when we add all of these things up, we're going to get 0, which means a plus b plus c plus d is equal to 0. Now, is that helpful at all? Let's find out. We have another equation that says 2a plus b plus c is equal to x, for example, right? We can write it as a plus a plus b plus c is equal to x. So from here, we could possibly do the following. For example, we know that a plus b plus c is equal to negative d, for example. I can isolate a plus b plus c from here and replace it with negative d. And here we get something like negative d. That gives us a minus d is equal to x. Great. Let's try the same thing on the second equation. This is b plus a plus b plus d. a plus b plus d is negative c. So this is b minus c is negative x. Let's see if this is going to help at all. And there's obviously so many ways to do this. c plus c plus d plus a is equal to negative b. That's 1 over x. So c minus b is 1 over x. So we're kind of getting two variables at a time. And this should give us d plus negative a. Right? bcd is negative 1 over x. So d minus a is negative 1 over x. Hmm. This is kind of interesting because a minus d and d minus a are supposed to be opposites. So we shouldn't be getting something like this. Maybe we're going to arrive at a contradiction. So anyway, I can try a different method maybe for this equation or system. Let's try something else. I'm going to erase everything that I did and start with another method because I didn't really like this one. Okay. So given these four equations, what are some of the things that we can do? Right? First of all, we know that a plus b plus c plus d is equal to 0. So that's a good thing to do or good thing to know. So maybe one thing we can do is, let's see, b plus c can be replaced. Or you know what I can do is I can try to eliminate some of these variables. For example, if I add the first two equations, I'm also getting 0. Let's check that out. 2a plus 3b plus c plus d is 0. And we also know that a plus b plus c plus d is 0, which means 3 times a plus b plus c plus d is 0. So from here we basically get that a plus b and 3 times a plus b are the same because they basically mean the same thing. So from here we get a plus b equals 0, which is cool. That also means c plus d is equal to 0. Does that mean everything is 0? Probably. But we can continue checking. For example, what did I do? 3a plus 3d. Okay, I added those two. Let's go ahead and add these two equations. 3c plus 3d plus a plus b is 0. But we already knew that, right? This is 0, this is 0. Okay, that doesn't help at all. Maybe try something else. Okay, let's see. We know that a plus b is 0, so maybe we can go ahead and use this first equation. Okay, a plus a plus b plus c is equal to x. We know that this is 0, so from here we get a plus c equals x. Great. And we know that a plus b is 0 and c plus d is 0. So I just need one more equation. Maybe I can get it from one of these. Let's see. I tried the 2a and maybe I can use this one. Can I? Okay, c plus c plus d plus a is 1 over x. Okay, we know that a plus c plus d is equal to negative b. So from here we get c minus b is 1 over x. Okay, we already knew that, right? Beforehand. But anyways, we got 4 variables and 4 equations. Hopefully this will work. I hope so. Now, I could probably... Let's see, what am I doing? I can try to eliminate c from here. Okay, let's go ahead and subtract these. d plus b is negative 1 over x. Okay. I was able to eliminate c. And from here I can't eliminate c, so that's kind of problematic. I don't know, I'm kind of getting stuck. Oh, okay. From here I can set the a's equal to each other. x minus c is 0 minus b, because they're both equal to a. And then I also know that... Okay, what does that mean? It just means that b plus c is equal to x. Hmm. b plus c is equal to x. a plus b is equal to 0. a plus c is equal to x. Now this means something. We can replace b with negative a from here. If we replace b with negative a, I get c minus a is equal to x. But a plus c is also equal to x. Which means a plus c is equal to c minus a. I'm probably taking the longest way possible. This means a is equal to 0. Great, this is helpful. If a is 0, then... Hmm. Okay. If a is 0, then c is x. Great. And if a is 0, b is also 0, because a plus b is 0. c plus d is 0, that means d is equal to negative x. Hmm. And c minus b... c minus b is 1 over x. This doesn't make sense at all, because this gives me x equals 1 over x, which is weird. But let me go ahead and check these values. What am I looking for? I'm looking for f of x, which is a. And a happens to be... Ta-da-da-da, 0. So does that mean f of x is identically equal to 0? Is this equation satisfied? Let's go back to the originals. If f of x is 0, 0 plus 0 plus 0 is not going to be x. So there's some issue. Maybe there are no solutions to this equation. And I just messed up, because I totally randomly made it up. Anyways. This brings us to the end of this video. 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 let me know what you think. Bye-bye.