Given a function f : D â R and a subset B â R, let fâ1(B) be the set of all points from the domain D that get mapped into B; that is, fâ1(B) = {x â D : f(x) â B}. This set is called the preimage of B. Required:
a. Let f(x)= x^2. If A is the closed interval [0, 4] and B is the closed interval [1â1] , find f^-1(A) and f^-1(B). Does f^-1(AâB) = f^-1(A) n f^-1(B) in this case? Does f^-1(A U B) f^-1(A) U f^-1(B)?

b. The good behavior of preimages demonstrated in (a) is completely general. Show that for an arbitrary function g : R âR, it is always true that g^-1( âB)= g^-1(A) â g^-1(B) and g^-1(A U B) =g^-1(A) U g^-1(B) for all sets A, B â R.