Function's Fun f(x^2−2016x)=f(x)⋅x+2016

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Discussion Overview

The discussion revolves around the functional equation f(x^2 - 2016x) = f(x) · x + 2016, with participants exploring the implications of this equation, particularly in relation to finding the value of f(2017). The scope includes mathematical reasoning and problem-solving related to functions.

Discussion Character

  • Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant presents the equation f(x^2 - 2016x) = f(x) · x + 2016 and asks for the value of f(2017).
  • Another participant questions whether "x^2 - 2016x" can be simplified to "2016x^3".
  • A third participant references a similar problem from another site, indicating that a good hint was provided there for determining f(2017).
  • A later reply suggests that f(x^2 - 2016x) equals f(2017) when x^2 - 2016x = 2017, leading to the factorization (x + 1)(x - 2017) = 0, which gives x = -1 or x = 2017.
  • This same reply proposes two scenarios for f(2017): either f(2017) = -1(f(-1)) + 2016 or f(2017) = 2017 f(2017) + 2016, leading to the conclusion that f(2017) could be -1.

Areas of Agreement / Disagreement

Participants express differing views on the simplification of the equation and the implications for finding f(2017). There is no consensus on the correct interpretation or solution to the problem.

Contextual Notes

The discussion includes assumptions about the function f and its properties, which are not fully defined. The implications of the factorization and the relationships between the values of f at different points remain unresolved.

cooltu
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f(x^2*2016x) = f(x)x+2016
Then f(2017) = ?
 
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Isn't "x^2*2016x" the same as "2016x^3"?
 
Thanks, skeeter. That sounds more reasonable.

f(x^2- 2016x)= f(2017) if x^2- 2016x= 2017. That is the same as x^2- 2016x- 2017= (x+ 1)(x- 2017)= 0 so either x= -1 or x= 2017.

So either f(2017)= -1(f(-1))+ 2016 or f(2017)= 2017 f(2017)+ 2016.

For the latter, -2016= 2016 f(2017) so f(2017)= -1.
 

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