Exercise 1.2.9. Given a function f: D R and a subset BC R, let -'(B)...

 

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Exercise 1.2.9. Given a function f: D → R and a subset BC R, let ƒ-'(B) be the set of all points from the domain D that get mapped into B; that is, f-1(B) = {x € D : f(x) E B}. This set is called the preimage of B. (a) Let f(x) = x². If A is the closed interval [0, 4] and B is the closed interval [-1, 1), find f-1(A) and f-1(B). Does f-1(An B) = f-'(A) n f-'(B) in this case? Does f-1(AU B) = f-'(A) U f-1(B)? %3D %3D (b) The good behavior of preimages demonstrated in (a) is completely general. Show that for an arbitrary function g : R → R, it is always true that g-1(An B) = g1(A) ng-1(B) and g-1(AU B) = g¯'(A) U g-1(B) for all sets A, BCR. %3D Exercise 1.2.9. Given a function f: D → R and a subset BC R, let ƒ-'(B) be the set of all points from the domain D that get mapped into B; that is, f-1(B) = {x € D : f(x) E B}. This set is called the preimage of B. (a) Let f(x) = x². If A is the closed interval [0, 4] and B is the closed interval [-1, 1), find f-1(A) and f-1(B). Does f-1(An B) = f-'(A) n f-'(B) in this case? Does f-1(AU B) = f-'(A) U f-1(B)? %3D %3D (b) The good behavior of preimages demonstrated in (a) is completely general. Show that for an arbitrary function g : R → R, it is always true that g-1(An B) = g1(A) ng-1(B) and g-1(AU B) = g¯'(A) U g-1(B) for all sets A, BCR. %3D

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