I came across this problem and it seems quite interesting, so, I'll present it to you guys.
Problem Statement:
If that looks really confusing, don't worry about it, we'll go through everything. Firstly, let us consider the left hand side of the equation. The first thing one should notice is that P is a function. You're probably not wondering, "Well, I've seen functions like P(x) before, but, what does P(P(x)) mean?", but, I'm going to tell you anyway, since some of you might be wondering exactly that.
Let me sidetrack for a moment here. Suppose you have a function f(x) = x^2. We know that
And so on. What if, we take the value of f(3) (i.e. 9) and stick back into f?
That can also be written as
Now, read carefully because this is where a lot of people have trouble, but, a smart individual like yourself should be able to figure it out in a snap!
Say, instead of f(3), we stick back in f the value f(x), so we can write that as
What if, we let y = f(x)? Since in the equation f(x) = x^2, x is just a placeholder (we can call it y or n or k or whatever the heck you want), we get:
And, substituting y = f(x), we get:
And, that's how functions inside functions work, this usually known as functional composition, mostly because mathematicians use big names to make themselves look clever (I've also noticed this with doctors) :P.
So, back to the problem at hand. The left hand side is basically taking P(x) (a polynomial) and sticking it into P(x) again. Now, that we have a rudimentary idea of how things are at the left, lets look at the right side of the equation.
That looks a lot less intimidating to me, basically, saying:
Now, this is where we actually get to attack the problem. What if, just like in our discussion of functional composition, we set
And, the equation transforms into:
And, since x and y are just placeholder variables,
So, that's it, we're done! That basically means that all equations that have x and an exponent will have the property mentioned in the problem, examples are:
Problem Statement:
If that looks really confusing, don't worry about it, we'll go through everything. Firstly, let us consider the left hand side of the equation. The first thing one should notice is that P is a function. You're probably not wondering, "Well, I've seen functions like P(x) before, but, what does P(P(x)) mean?", but, I'm going to tell you anyway, since some of you might be wondering exactly that.
Let me sidetrack for a moment here. Suppose you have a function f(x) = x^2. We know that
And so on. What if, we take the value of f(3) (i.e. 9) and stick back into f?
That can also be written as
Now, read carefully because this is where a lot of people have trouble, but, a smart individual like yourself should be able to figure it out in a snap!
Say, instead of f(3), we stick back in f the value f(x), so we can write that as
What if, we let y = f(x)? Since in the equation f(x) = x^2, x is just a placeholder (we can call it y or n or k or whatever the heck you want), we get:
And, substituting y = f(x), we get:
And, that's how functions inside functions work, this usually known as functional composition, mostly because mathematicians use big names to make themselves look clever (I've also noticed this with doctors) :P.
So, back to the problem at hand. The left hand side is basically taking P(x) (a polynomial) and sticking it into P(x) again. Now, that we have a rudimentary idea of how things are at the left, lets look at the right side of the equation.
That looks a lot less intimidating to me, basically, saying:
Now, this is where we actually get to attack the problem. What if, just like in our discussion of functional composition, we set
And, the equation transforms into:
And, since x and y are just placeholder variables,
So, that's it, we're done! That basically means that all equations that have x and an exponent will have the property mentioned in the problem, examples are:
Note: For those of you that want crazy rigor, you may notice that y is actually a function of x and not a placeholder variable, but, the idea in a solution that doesn't rely on x isn't much different from this.
Basically, the technique we used was very simple, since there were P(x)'s on both sides of the equation, we used a special technique (by setting y=P(x)) to sort of get rid of some of the P(x)'s.
This problem was a small taste of quite large field known as functional equations. If you want to find out some more about this subject, I would suggest this as a starting point, and this after that, and this book, if you're really interested.
If you liked reading this, and know a bit of computer programming, I suggest you head on over to http://poincare101.blogspot.com/, another blog of mine that deals with computer programming, and more advanced (usually pure) mathematics.
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