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8. (8pts) Find the range of these functions. Give your answer as a simple/short sentence using plain English. For small ranges you can use roster notation. The following solution to (a): S = {2x | x € Z} despite being correct, will receive 0 points. Refer to part (a) for a sample solution. Recall that N is the set of all natural numbers {0, 1, 2, 3,...}, and Z+ is the set of positive integers {1,2,3,...}. (a) f: Z→R, f(x) = 2x. Solution: The range is the set of even integers. (b) f: NR, f(x) = 2x. (c) f: RR, f(x) = 2x. (d) f: NR, f(x) = last digit of 2x. (e) f: NxZ+→R, f(x, y) = x/y.

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