Suppose f : N?N satisestherecurrencerelation f(n + 1) (f(n) 2 iff(n)iseven 3f(n)+ 1 if f(n)isodd . Notethatwiththeinitialconditionf(0) 1,thevaluesofthefunction are: f(1) 4, f(2) 2, f(3) 1, f(4) 4,and so on, the images cyclingthroughthosethreenumbers. Thus fisNOTinjective(andalso certainlynotsurjective).Mightitbeunderotherinitialconditions?3 (a) If fsatisestheinitialcondition f(0) 5,is f injective? Explainwhyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage.(b) If f satisestheinitialcondition f(0) 3,is f injective? Explainwhyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage.(c) If f satisestheinitialcondition f(0) 27,thenitturnsoutthatf(105) 10 and no two numbers less than 105 have the same image.Could f beinjective? Explain. (d) Prove that no matter what initialcondition you choose, the functioncannotbesurjective.
