# Can polynomial many-to-one reduction be done to a specific problem instance?

Let's say I reduce the problem $$A \in L$$ to $$B \in K$$ , with a function $$f: \Sigma^{*} \rightarrow \Gamma^{*}$$ such that $$w \in L \Leftrightarrow f(w) \in K$$ . I know if I want to solve $$A$$, given some polynomial time algorithm for $$B$$, I just have to transform $$A$$ to $$B$$ and solve $$B$$. So it can be thought as:

The reduction must be done from arbitrary instance of $$A$$ to a legal instance of $$B$$

My question is, do I have to reduce to arbitrary instance of $$B$$ or some instance of $$B$$? I.e. reduction from TQBNF to Generalized Geography is done to some valid graph instance, but there exist many more valid instances of Generalized Geography.

• So $f$ can be described as a function that can be injective and can be surjective? – carpenter Dec 12 '19 at 16:56