Let N denote the set of all natural numbers, and Z denote the set of all integers. Consider the functions f: N Z and g: Z N defined by f(n)= cases (n+1) / 2 & if n is odd \\ (4-n) / 2 & if n is even cases And g(n)= \ array cc 3+2 n & if n 0 \\ -2 n & if n <0 array . Define (g f)(n)=g(f(n)) for all n N , and (f g)(n)=f(g(n)) for all n Z . Then which of the following statements is (are) TRUE?
D. f is NOT one-one but f is onto
aligned & f(n)= cases (n+1) / 2 & if n is odd \\ (4-n) / 2 & if n is even cases \\ & f(n)=\ (1,1),(2,1),(3,2),(4,0),(5,3),(6,-1), \ aligned f ( n ) is many one and onto function aligned & g ( n )= \ array cc 3+2 n & if n 0 \\ -2 n & if n <0 array . \\ & g ( n )=\ (-3,6),(-2,4),(-1,2),(0,3),(1,5),(2,7),(3,9),(4,15), \ aligned g ( n ) is one-one and into function f ( ~g ( n ))=2+ n , n ~N fog is one-one and into g(f(n))= cases 4+n & if n is odd natural number \\ 7-n & if n=2,4 \\ n-4 & if n is even natural number and n 6 cases g(f(2))=g(f(1))=5 gof is many one and into
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