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

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The discussion revolves around the functional equation f(x^2 - 2016x) = f(x)·x + 2016, with a focus on finding f(2017). It is established that x^2 - 2016x = 2017 leads to two possible values for x: -1 and 2017. Substituting these values results in two equations for f(2017), one involving f(-1) and the other a direct calculation. The latter simplifies to f(2017) = -1, concluding that f(2017) equals -1. The problem highlights the importance of correctly interpreting the functional equation to derive the solution.
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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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