# Find possible states in an automaton from a given input sequence

I have an automaton (specifically a nondeterministic finite automaton, NFA) and I am trying to determine the possible states that the automata could be in, given a specific sequence of input symbols (not necessarily starting at the start state).

For example, consider the following NFA: Given the input sequence $$(0,1)$$ we know we could be at either $$q_0$$ or $$q_3$$. If the input sequence is $$(0,0,0)$$ we could be at any of the states $$\{q_0,q_1,q_2\}$$.

I'm curious if this problem has been studied in automata theory -- does it have a name?

In view of the deterministic automaton, there is a function $$f(q_0,w)$$, which gives for the (unique) initial state $$q_0$$ and each sequence $$w$$ the ending state after reading the input.