# Odd and Even Functions for equations I don't know the exponents of

I'm solving a problem to which the final obstacle (I think) is knowing if this function is even or odd:

Given $$xQ(x+2018)=(x-2018)Q(x)$$ and that $$Q(1)=1$$, what is $$Q(2017)$$?

My best attempt so far is plugging in $$x =-1$$ to which I got $$(-1)Q(-1+2018)=(-1-2018)Q(-1)$$ $$Q(2017)=(2019)Q(-1)$$ I can simply conclude $$Q(2017)=2019$$ if I can prove $$Q(-1)=Q(1)$$ which brings me to the question how can I prove that $$xQ(x+2018)=(x-2018)Q(x)$$ is an even function? I've only been used to identifying odd and even functions for the usual polynomial/rational functions with exponents and all.

I would really appreciate the help. Thank you!

• @robgimarino: As currently stated, there's not enough information. Are you sure you've stated the problem fully? – quasi Nov 2 '18 at 5:46
• I've stated the problem exactly as reflected in the paper – robgimarino Nov 2 '18 at 6:59
• What paper? A homework sheet? Can you post an image or a link? – quasi Nov 2 '18 at 7:21
• @Robgimarino I can disprove your claim. Try to find f(-2017) which you will get 2017/4035. If f(x)=f(-x) for all x,then f(-2017)=f(2017) proving that your claim f(1)=f(-1) is wrong. – Jasmine Nov 3 '18 at 7:45
• I must be looking at the problem wrongly, I see. Thanks for letting me know :) I'll try to look at it in another way – robgimarino Nov 3 '18 at 9:53