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) = 0, f (n) = 2f (n − 2) for n ≥ 1
b) f (0) = 1, f (n) = f (n − 1) − 1 for n ≥ 1
c) f (0) = 2, f (1) = 3, f (n) = f (n − 1) − 1 for n ≥ 2
d) f (0) = 1, f (1) = 2, f (n) = 2f (n − 2) for n ≥ 2
e) f (0) = 1, f (n) = 3f (n − 1) if n is odd and n ≥ 1 and f (n) = 9f (n − 2) if n is even and n ≥ 2.