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[answered] Let h: Z>Z be the function defined by h(x)=2x+1 for


Let ?h:?Z?>Z?be the function defined by?h(x)=2x+1 for all?x belongs to?Z.

(a) Prove that h is one-to-one but not onto Z.
(b) Is there a function g : Z ->Z so that g ??h is onto Z?
(c) For positive integers n, de ne the functions h^n : Z ->?Z by putting h^1 = h and for integers k>=? 2, h^k?= h?h^k-1. Compute h^2 (x), h^3 (x), h^4 (x), and h^5 (x). From these, guess an explicit formula for h^n (x) for all integers n >=?1.
(d) Prove by induction on n that your guess in part (c) is correct.


 


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