Let $f(x) = 1/(1-x)$. Show that if x ≠ 0,1, then $x$ is a period 3 point.
What I did was I took the 3rd iterate $f^3(x) = f(f(f(x)=x$, ie I took $ -\frac{1-\frac{1}{1-x}}{\frac{1}{1-x}}$ and when simplified you get $x$.
Graphically it creates a closed-loop as long as it is within (0,1). but how can I prove that $x$ is period 3 point. Is it possible to solve this algebraically or do I need to plug in numbers ?
You have shown that $f^3(x)=x$ for $x\ne0,1$. Therefore the function either has period $1$ or period $3$.
If it had period $1$ then $x=\frac{1}{1-x}$ which has no real solutions and therefore it has period $3$.