searching for $f(x)$ when knowing $f(2^{2x})$ I have a function that works for powers of 2
This is only for INTEGERS
is there a way to calculate any integer x?
$$f(2^{2x})=\frac{4^x+2}3.$$
$f(x)$=?
here are the first 100 values of f(x)
1, 2, 2, 2, 2, 4, 4, 6, 6, 8, 8, 6, 6, 8, 8, 6, 6, 8, 8, 6, 6, 8, 8, \ 14, 14, 16, 16, 14, 14, 16, 16, 22, 22, 24, 24, 22, 22, 24, 24, 30, \ 30, 32, 32, 30, 30, 32, 32, 22, 22, 24, 24, 22, 22, 24, 24, 30, 30, \ 32, 32, 30, 30, 32, 32, 22, 22, 24, 24, 22, 22, 24, 24, 30, 30, 32, \ 32, 30, 30, 32, 32, 22, 22, 24, 24, 22, 22, 24, 24, 30, 30, 32, 32, \ 30, 30, 32, 32, 54, 54, 56, 56, 54 is there a pattern to calculate any f(x)?
thanx

In general if we have,
$$f(g(x))=h(x)$$
And if there exists an inverse to $g$. Then let $u=g(x)$. This implies $g^{-1}(u)=x$ where this notation refers to the inverse of $g$. Substituting  in we have,
$$f(u)=h(g^{-1}(u))$$
Which means,
$$f(x)=h(g^{-1}(x))$$
