a piecewise function definition

1. Understand the function definition: The problem defines a piecewise function $f(n)$. This means the output of the function depends on the input value of $n$. There are two rules: one for when $n$ is an even number, and another for when $n$ is an odd number.

1. Rule for even n: If the input $n$ is an even number, the function's output is calculated by dividing $n$ by 2. This can be written as $f(n) = \frac{n}{2}$.

1. Rule for odd n: If the input $n$ is an odd number, the function's output is calculated by multiplying $n$ by 3 and then adding 1. This can be written as $f(n) = 3n + 1$.

4. Summary of the function: Combining these rules, the function $f(n)$ is defined as:
$$f(n) = \begin{cases} \frac{n}{2} & \text{if } n \text{ is even} \\ 3n + 1 & \text{if } n \text{ is odd} \end{cases}$$