Question

Let \( \mathbb{N} \) be the set of natural numbers and \( f : \mathbb{N} \to \mathbb{N} \) be defined by 

Let \( f^n(x) \) denote the \( n \)-fold composition of \( f(x) \). What is the smallest integer \( n \) such that \( f^n(13) = 1 \)? 
 

Hint:

For these types of problems, keep track of each step and apply the function according to whether the number is odd or even.

Answer 1

Correct Answer: 9

Step 1: Understanding the function.
The function \( f(x) \) operates differently depending on whether \( x \) is even or odd. If \( x \) is even, it is halved; if odd, it is replaced by \( 3x + 1 \). We need to find the smallest \( n \) such that after \( n \) applications of \( f \), the value reaches 1 starting from 13.
Step 2: Applying the function to 13.
- \( f(13) = 3 \times 13 + 1 = 40 \) (odd)
- \( f(40) = 40/2 = 20 \) (even)
- \( f(20) = 20/2 = 10 \) (even)
- \( f(10) = 10/2 = 5 \) (even)
- \( f(5) = 3 \times 5 + 1 = 16 \) (odd)
- \( f(16) = 16/2 = 8 \) (even)
- \( f(8) = 8/2 = 4 \) (even)
- \( f(4) = 4/2 = 2 \) (even)
- \( f(2) = 2/2 = 1 \) (even)
Thus, after 6 steps, we reach \( f^6(13) = 1 \).
Step 3: Conclusion.
The smallest integer \( n \) such that \( f^n(13) = 1 \) is 6.