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

[cooltu]

f(x^2*2016x) = f(x)x+2016
Then f(2017) = ?

 

[HallsofIvy]

Isn't "x^2*2016x" the same as "2016x^3"?

 

[skeeter]

similar problem from another site had $f(x^2-2016x) = f(x) \cdot x + 2016$

https://mathhelpforum.com/threads/functions-problem.285171/#post-954658

very good hint was given to determine $f(2017)$

 

[HallsofIvy]

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.