91Ó°ÊÓ

A one-to-one function (also called an injective function) is a function that maps distinct elements in its domain to distinct elements in its range. The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs. In this exercise, the domain is the set of integers, and the range is the set of positive integers.

A possible function that meets the requirements is a piecewise function that maps even integers to positive even integers and odd integers to positive odd integers. Let's define the function \(f(x)\) as follows: \[f(x) = \begin{cases} x/2, & \text{if } x \text{ is even}\\ -(x-1)/2, & \text{if } x \text{ is odd} \end{cases}\]

To prove that \(f(x)\) is one-to-one, we need to show that if \(x_1 \neq x_2\), then \(f(x_1) \neq f(x_2)\). In other words, we need to show that distinct integers map to distinct positive integers. Let's suppose \(x_1 \neq x_2\) and consider the following cases: Case 1: Both \(x_1\) and \(x_2\) are even. In this case, \(f(x_1) = x_1/2\) and \(f(x_2) = x_2/2\). Since \(x_1 \neq x_2\), it follows that \(f(x_1) \neq f(x_2)\). Case 2: Both \(x_1\) and \(x_2\) are odd. In this case, \(f(x_1) = -(x_1-1)/2\) and \(f(x_2) = -(x_2-1)/2\). Since \(x_1 \neq x_2\), it follows that \(f(x_1) \neq f(x_2)\). Case 3: One of the inputs (\(x_1\) or \(x_2\)) is even, and the other is odd. Without loss of generality, let's assume that \(x_1\) is even and \(x_2\) is odd. In this case, \(f(x_1) = x_1/2 > 0\) and \(f(x_2) = -(x_2-1)/2 < 0\). Since \(f(x_1)\) is positive and \(f(x_2)\) is negative, it follows that \(f(x_1) \neq f(x_2)\). In all three cases, we can conclude that as long as \(x_1 \neq x_2\), \(f(x_1) \neq f(x_2)\), so the function \(f(x)\) is indeed one-to-one.

We defined the function \(f(x)\) in such a way that it maps each integer to a unique positive integer. The function is defined for all integers, and due to the one-to-one property we have verified, the range will only include distinct positive integers. Thus, the domain and range are the set of integers and the set of positive integers, respectively. So, the example of a one-to-one function whose domain is the set of integers and whose range is the set of positive integers is given by \[f(x) = \begin{cases} x/2, & \text{if } x \text{ is even}\\ -(x-1)/2, & \text{if } x \text{ is odd} \end{cases}\]