I try to explain you another way, with your math language:
1) you know f(2) = 0, f(5)=8, f(8)=0
2) f(x) is continue function, so there is a A, 2<A<5, and f(A)=2; and a B, 5<B<8, f(B)=2
since 2 is between 0 and 8
3) you will get f(f(A))=f(f(B))=f(2) = 0, and f(f(5))=f(8)=0
so she said f(f(x)) would have 3 roots (no use to calculate A and B, as you did)
give you the same explanation,