The discussion revolves around a polynomial function f(x) that satisfies a specific functional equation involving two variables, x and y. The original poster states that f(2) equals 5 and seeks to find the value of f(f(2)). Participants explore the implications of the functional equation and the nature of polynomial functions.
The discussion is active, with participants sharing various approaches and insights. Some have proposed specific substitutions to simplify the problem, while others are questioning the degree of the polynomial and the nature of its roots. There is an ongoing exploration of the implications of the functional equation, with no explicit consensus reached yet.
Participants note the importance of the condition f(2)=5 and how it relates to the functional equation. There is a recognition that the degree of the polynomial and the nature of its roots are critical to understanding the problem, but these aspects remain under discussion.
f(x)=g(x)+1haruspex said:Exactly. So, what is the general form of g(x), and thus, the general form of f(x)?
Well, yes, that's the relationship between f(x) and g(x), but what is the general form of g(x)? We have shown that all its roots are 0, right? So what does that polynomial look like?utkarshakash said:f(x)=g(x)+1
haruspex said:Well, yes, that's the relationship between f(x) and g(x), but what is the general form of g(x)? We have shown that all its roots are 0, right? So what does that polynomial look like?
Indeed I did, but from your post #29 it seemed like you'd not been reading all those exchanges, perhaps because you wanted to figure it for yourself with a few hints.utkarshakash said:You have already stated that in an earlier post
haruspex said:Indeed I did, but from your post #29 it seemed like you'd not been reading all those exchanges, perhaps because you wanted to figure it for yourself with a few hints.
So, do you understand why g(x) = axk for some a and k? Do you understand how to determine a from the equation for g(), and then the value of k from the given datapoints?
utkarshakash said:I do not know how to derive a but assuming a=1, I can find k.
In your post 27 you wrote, correctly,utkarshakash said:I do not know how to derive a but assuming a=1, I can find k.
Substitute g(x) = axk in there.g(x)g(y)=g(xy)
haruspex said:In your post 27 you wrote, correctly,
Substitute g(x) = axk in there.