TITLE: Composition of functions and number of solutions
QUESTION [5 upvotes]: If $f(x)= 4x(1-x)$, if  $x \in [0,1]$.
find number of solutions of equation $$f(f(f(x)))={x\over 3}$$
method: i tried finding the compostion but it is too lenghty. 
also i tried put $x= \sin ^2 a$ but no success.

REPLY [5 votes]: You have an excellent idea!
Putting $x=\sin^2a$ we have $$f(\sin^2 a)=\sin^22a\implies f\circ f=\sin^24a\implies f\circ f\circ f=\sin^28a$$ so we need to solve $$3\sin^28a-\sin^2a=0$$