jmc

Question

A function f from the integers to the integers is defined as follows: f(n) = arraycl n + 3 & if n is odd, n/2 & if n is even. array .Suppose k is odd and f(f(f(k))) = 27. Find k.

Accepted Answer

Since k is odd, f(k) = k + 3. Then k + 3 is even, so f(k + 3) = k + 32.If k + 32 is odd, then f ( k + 32 ) = k + 32 + 3 = 27.This leads to k = 45. But f(f(f(45))) = f(f(48)) = f(24) = 12, so k + 32 must be even. Then f ( k + 32 ) = k + 34 = 27.This leads to k = 105. Checking, we find f(f(f(105))) = f(f(108)) = f(54) = 27. Therefore, k = 105.