



 
Hi Nahla, I'll give you some hints also and hopefully they will help. Start with the first condition and see if $993$ can be the result of $f \circ f(n)$ because if is is than $f(993) = f \circ f \circ f(n).$ If $f \circ (n) = 993$ then $4n  3 = 993$ and $n = 249.$ Hence $f \circ f(249) = 993.$ What about $249?$ If $f \circ (n) = 249$ then $4n  3 = 249$ and $n = 63.$ Hence $f \circ f(63) = 249.$ But $63$ is one less than a power of $2$ and that makes me think of the second condition, $f(2^n) = 2^{n+1} 1.$ A fact that is useful is that $f \circ f \circ f(n) = f \circ f \left(f(n) \right) = f \left( f \circ f(n) \right).$ Penny  


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