How do I determine periodicity of a function through a system of functional equations?

Question

I was given these equations, $f(k+x) = f(k-x)$, $f(2k+x)= -f(2k-x)$ . k is assumed a constant. I was asked to comment whether $f(x)$ is even or odd. By solving I came to the equation, $f(-x)=-f(x)$,which states that $f(x)$ is an odd function. But how do I comment on its periodic nature?

Answer 1

Since an odd function(you have proved), implies$$ -f(2k-x)=f(x-2k)$$ So $$f(x-2k)=f(x+2k)$$ so period is $4k$.