If $f$ is a function of integers, define $f(L)$ as:
$$ f(L) = \{w \mid \text{for some } x, \text{ where } |x| = f(|w|), \, wx \in L \}. $$
This means $f(L)$ is the set of strings $w$ such that there exists some string $x$ of length $f(|w|)$ for which the concatenation $wx$ belongs to $L$. For example, the operation ( $ \text{half}(L) $ ) corresponds to the function $f(n) = n$, as this ensures that $ |x| = |w| $.
You are asked to prove that if $L$ is a regular language, then $f(L)$ is also a regular language, for the following functions $f(n)$:
$f(n) = 2n$ (i.e., the amount of the string we take corresponds to the first third of the string).
$f(n) = n^2$ (i.e., the length of the substring we take equals the square root of the length of the remaining part).
$f(n) = 2^n$ (i.e., the length of the substring we take equals the logarithm of the length of the remaining part).
I have resolved the first question, but confused by the other two questions.
