Relations & Functions
Iterated function; tracing back
Grade Class 12
Question:
A function $f:\mathbb{I}\to\mathbb{I}$ is: $f(n)=n+3$ (odd $n$), $f(n)=n/2$ (even $n$). $k$ is odd and $f(f(f(k)))=27$. Then sum of digits of $k$ is
Step-by-Step Solution
Key Concept: Work backwards from $27$: $f^{-1}(27)\to f^{-1}\to k$. Even preimage of $27$: $54$; odd preimage of $54$: not possible (51 is odd, $f(51)=54$ ✓). Then $f(k)=51$ with $k$ odd: $k+3=51\Rightarrow k=48$ (even — invalid). Try $108$: $f(k)=108\Rightarrow k=105$ (odd ✓).
$k=105$, sum of digits $=6$.
Correct Answer: 2