How to solve a functional equation on the AIME

Hey, blackpenredpen. Can you make a solution video on this problem? This is AIME 2012 problem number 14. Though it has solutions online, I couldn't understand them. Hope you read this. Thank you. Okay, let's have a look. So, we have a functional equation. Given that f of x * f of 2x squared is equal to f of 2x cubed + x. That's true for all x. And we know f of 0 is equal to 1. f of 2 + f of 3 is equal to 125. And the goal is to find out the value for f of 5. Hmm. So, this is how I did it. Firstly, I think the biggest hint is this right here. 2x cubed + x. Here we can factor on x. So, this equation is the same as f of x * f of 2x squared, and that is equal to f of factoring on x, we have 2x squared + 1. Like this. Now, have a look. This inside times that inside is very similar to this inside because we have the x and also the 2x squared from here and here. It's just that we have that + 1 right here in that parentheses. Okay. What do we do next though? Well, usually when we are dealing with a functional equation, we want to start with some guess and hope for the best. And of course, you want to start with a good guess. I don't think you want to use log function because you have a log function times a log function, you cannot multiply the inside. Well, you don't want to use exponential functions either because exponential function times exponential function, you will have to add the exponents, right? You inside. I want to multiply the inside. So, I will start with a power function. So, right here this is the general form that I'm going to try. Let's say we have f of x. Let's say this is equal to some power. I don't know what power it is yet, so I'll just put down n. So, right here let's say we have the input x. So, that's my general form of the function, okay? And if I have this, f of x will be what? It's just this part, parentheses to the nth power, and then we have the x right here. So, this is the input. I'll put it here. And then this right here, we multiply by this input, which is 2x squared, and then nth power, and that will be equal to this thing. And you see, I will just put it here. x 2x squared + 1 like that. And as you can see, when we have a power function times a power function, we really just multiply the inside. So, the left-hand side becomes like this. And that will be set as that. And right now, we can just multiply the x and that, but of course, this is not correct. So, we have to make more change. Let's take a look at the next information. We have f of 0 is equal to 1. I don't want to put a + 1 on the outside because otherwise, I cannot multiply the insides like this, right? So, let me try + 1 right here. So, that means when we have f of x, we will have x + 1 to the nth power. And this will be 2x squared and then + 1 to the nth power. This is right here, but I still have to put on the + 1 because we have that right here. Okay. Right here, let me just try to see if this is actually going to work. If it is working, then we have the function, and we can just continue. If not, we will have to make more changes. Okay, so let's see. Multiply the inside, the first one is just going to be x + 1, and then the second one is 2x squared + 1, and that will be equal to this, right? So, it's x * 2x squared + 1, and then + 1. I don't want to distribute this because if you just focus on the inside, I can take the x times the first whole thing here, so I get x parentheses 2x squared + 1. That is really good, right? And then 1 * 1 will be a + 1. That is really good. However, once we do the x times the parentheses right here, I will have to go with the 1 multiplied by this, which is + 2x squared. And then we have that + 1. Oh my goodness. Almost, right? Almost. Hmm. You see that? We have that 2x squared, so what do we do?

do? Well, right here, we may have to make more changes. And I think I'm just going to work with the power here. Since I have an unknown power already, so I'm going to just kind of guess a power and see if I'm lucky enough or not. So, what I'm going to next is I will try x to the second power. So, if I make that change, f of x will be this thing, but the x will be to the second power. And this thing right here will also power. Okay, and then this right here, we will just have to change to x squared. Good, but this right here will become this thing squared, which is Let me just rewrite everything again. So, we have x squared + 1 times this thing, which is I'm just going to write the yes. Now, let's expand it. So, it's 4x to the fourth power, and then + 1. And then to the nth power. And let's make it equal to this. And right here, I will distribute the power, so it's x squared times this thing, which is 2x squared + 1, and then squared, and then + 1 to the nth power. And let's see if I take x squared times this thing, like this whole parentheses, I get x squared times 4x squared 4x to the fourth power + 1. Okay. That is kind of similar to this because if you expand this, you have x squared, and the first term of this expansion is precisely 4x to the fourth power. Good. And the last term right here is the + 1. Good. But in the middle here, what do we have? we have 2 times this and that, so we will have to have the + 4x squared. So, so far, they are not the same, but don't worry because once we take this, multiply with that, we move to the 1, and then we multiply with this. How does that look? It looks really good because this times that is + 4x to the fourth power, and then lastly, we have that + 1. nth power, and then this thing is from here, and then we have the + 1 raised to the nth power. Now, ladies and gentlemen, you see that if you Oh, if distribute, this thing is 4x to the sixth power, and then + 4x to the fourth power. Yes, and then + x squared, and then + 1. That's exactly what we have right here. So, this and that is 4x to the sixth power, and we can add that, and + x squared, and then + 1. And then both of them are to the nth power. Okay, so this right here is the form that we know for the function. But we still don't know what n is though. That's where we have to use this information. So, let's go ahead and continue. So, now we know this is the form of our function. We still don't know what n is yet though, but it's okay. Let's see f of 2. This is just going to be 2 squared + 1, which is 5, and then raised to the nth power. And then similarly, f of 3 squared + 1 is 10, and then raised to the nth power. So, when we have this plus that is equal to 125, that means 5 to the n + 10 to the n is equal to 125. And for this equation, it's not so bad because if n is equal to 2, we get 100, and that happens to be 5 to the second power, which is 125. It matches that perfectly. So, n is equal to 2. So, all that tells us our function is just x squared + 1 squared. Now, we just have to figure out what f of 5 is. So, that will be 5 squared + 1 squared, which will be 26 squared, and work that out, 676. That was it.