Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f (n) when n is a nonnegative integer and prove that your formula is valid. a) f(0) = 1, f(n) = −f(n − 1) for n ≥ 1 b) f(0) = 1, f(1) = 0, f(2) = 2, f(n) = 2f(n − 3) for n ≥ 3 c) f(0) = 0, f(1) = 1, f(n) = 2f(n + 1) for n ≥ 2 d) f(0) = 0, f(1) = 1, f(n) = 2f(n − 1) for n ≥ 1 e) f(0) = 2, f(n) = f(n − 1) if n is odd and n ≥ 1 and f(n) = 2f(n − 2) if n ≥ 2

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