g(x)+g(-x)=0
f(x+k)+f(-x+k)=0
(x+k-1)/(x+k+3)*(-x+k-1)/(-x+k+3)=1
(k-1)^2-x^2=(k+3)^2-x^2
-2k+1=6k+9
8k=-8
k=-1