I am struggling with the following problem
Let $f:\mathbb{N} \to \mathbb{N}$ be a function on natural numbers and say $L$ is some language.
Define $L_f = \left \{ x \mid \exists y \in \Sigma^*: |y| = f(|x|) , x \cdot y \in L \right \}$
If $L$ is regular and $f$ is $x \mapsto x^2$ and $g$ is $x \mapsto 2^x$, show that $L_f$ and $L_g$ are regular.
I have solved the FirstHalf problem in which the function is $x \mapsto x$, using the technque of placing pebbles.
I understand that this problem would require some similar idea, probably keeping track of the states in two concurrent automata, using products of Boolean Transition Matrices, but I am unable to fill in the details.
