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The function you've provided is a piecewise function that is related to the Collatz conjecture. Here's the detailed breakdown of the function \( f(n) \):

\[
f(n) =
\begin{cases} 
\frac{n}{2} & \text{if } n \text{ is even} \\
3n + 1 & \text{if } n \text{ is odd}
\end{cases}
\]

### Explanation:
- **If \( n \) is even**, the function outputs \( \frac{n}{2} \), meaning you simply divide \( n \) by 2.
- **If \( n \) is odd**, the function outputs \( 3n + 1 \), meaning you multiply \( n \) by 3 and add 1.

This is precisely the rule used in the Collatz conjecture, which hypothesizes that by iteratively applying this function, starting from any positive integer \( n \), you will eventually reach the number 1.

Would you like further details or have any specific questions about this function?

Here are five related questions you might explore:
1. What is the Collatz conjecture and why is it significant?
2. Can you provide an example where \( n \) is even and odd to demonstrate how this function works?
3. What happens when you repeatedly apply this function starting with a large number?
4. Are there any known counterexamples to the Collatz conjecture?
5. How can this function be visualized graphically?

**Tip**: When applying piecewise functions, always check the condition (even or odd) carefully before proceeding with the calculation!

f(n) = { n/2 if n is even, 3n + 1 if n is odd }

Discover how to solve the piecewise function f(n) = { n/2 if n is even, 3n + 1 if n is odd } and understand its relation to the famous Collatz Conjecture. Suitable for students in Grades 10-12.

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