I know to find a period-n point for a discrete dynamical function you solve $f^n(x)=x$.
However, in my textbook, one example is of the odd function, $$f(x)=x^3-\frac{5}{4}x$$ and it says because odd functions have the property $$f(-x)=-f(x)$$ we can find the period-2 point which satisfies $$x_2=f(x_1)=-x_1$$ and $$x_1=f(x_2)=-x_2$$ and thus, solve $$f(x)=-x$$
I don't understand why this is true.