91Ó°ÊÓ

Let's define a function \(f(n)\), where \(n\) is a positive integer. We want our function to output an integer for each positive integer input.

We can use the following rule for our function \(f(n)\): \[f(n) = \begin{cases} \frac{n-1}{2} & \text{if } n \text{ is odd} \\ -\frac{n}{2} & \text{if } n \text{ is even} \end{cases}\] This function takes a positive integer \(n\) and outputs an integer, with odd inputs resulting in positive integers and even inputs resulting in negative integers.

Let's verify that the domain and range are as desired. Domain: The domain is the set of positive integers. Since the function has a defined output for all positive integer values of \(n\) (either odd or even), the domain is indeed the set of positive integers. Range: We want the range to be the set of all integers. For odd values of \(n\), \(f(n) = \frac{n-1}{2}\) generates positive integers: - \(f(1) = 0\) - \(f(3) = 1\) - \(f(5) = 2\) - \(f(7) = 3\) - ... For even values of \(n\), \(f(n) = -\frac{n}{2}\) generates negative integers: - \(f(2) = -1\) - \(f(4) = -2\) - \(f(6) = -3\) - \(f(8) = -4\) - ... Therefore, the range of function \(f(n)\) is the set of all integers, as desired. The example of the function with the domain being the set of positive integers and the range being the set of integers is: \[f(n) = \begin{cases} \frac{n-1}{2} & \text{if } n \text{ is odd} \\ -\frac{n}{2} & \text{if } n \text{ is even} \end{cases}\]