6 DUE: MONDAY, AUGUST 28 AT 10:00 PM (1) Prove that if f and g are functions and Dom(f ) = Dom(g) and f (x) = g(x) for all x Dom(f ), then f = g. (2) Prove that if f : A B and Rng(f ) = C and f 1 is a function, then f f 1 = IC . (3) Let h : A B and g : C D, and let E = A C. Prove that h g is a function from A C to B D if and only if h|E = g|E . Note: If you want to use theorem 4.2.5, you should prove it first. But it is not necessary to use this theorem. (4) Let A be the interval (1, ) and let f be the relation on A given by \u001b \u001a x . f = (x, y) A A : y = x1 (a) Prove that f is a function from A to A. (b) Determine whether f : A A is surjective, and prove your answer. (c) Determine whether f is injective, and prove your answer. (5) Define functions f : R R and g : R R as follows: ( x + 4 if x 2 2 x if x 1 g(x) = x f (x) = 1 if 2 < x < 2 if x > 1 x x 4 if x 2 (a) Prove that f is injective but not surjective. (b) Prove that g is surjective but not injective