Other Integer Bases (Binary, Ternary, Etc.)

(I am pretty sure this is an AMC or AIME problem, but I can't find the original source.)

Consider the following function, defined for $x\in [0, 1):$ $$f(x) = 2x (0 \le x \lt 1/2)$$ $$f(x) = 2x – 1 (1/2 \le x \lt 1)$$ For how many values of $x$ does $f^5(x) = x$?

(As a bonus, can you find all of the values $x$ such that $f^5(x) = x$?)