Under what condition does $f(f^{-1}(f(A))) = f(A)$? Basic question regarding function.
Let $f: X \to Y$, then for what $f$ does $f(f^{-1}(f(A))) = f(A)$? hold?
Obviously this relationship holds when $f$ is a bijection. 
This does not hold when $f$ is pure surjection because the inverse does not exist.
Does this also hold when $f$ is an pure injection? I think so. 

When $f$ is an injection function then $f^{-1}(f(A))=A$, as a result
$$f(f^{-1}(f(A))) = f(A)$$
