of what subset is a combination function, a characteristic function?

Question

if $$f:A→B$$ and $$χ_E $$ is the characteristic function of $$E⊂B$$ of what subset of A, is $$χ_E\circ f $$ the characteristics function?

I know the characteristic function for all x in E, equals 1 and for any other x, it equals zero. I do not know how different this question is. Honestly, I really do not get the concept of this question. I would appreciate any help.

Answer 1

It is the characteristic function of $f^{-1}(E)$ since $\chi_E(f(x))=1$ if $f(x) \in E$ or $x \in f^{-1}(E)$ and $0$ otherwise.