- Define the domain as the set of all first components (x-values) from the given ordered pairs.
- Extract each x-coordinate from the function \(f(x) = \{(-16, 13), (-11, 82), (22, 48), (24, 26), (27, 31), (47, 98)\}\).
- Compile these values into a set to represent the domain.
- State the final domain as \(\boxed{\{-16, -11, 22, 24, 27, 47\}}\).
Explanation
- Defining the Domain
A function, in the context of a set of ordered pairs, is a relation that assigns each input (the first coordinate) to exactly one output (the second coordinate). The domain of such a function is defined as the set of all possible input values, which corresponds to the collection of all first components from the given ordered pairs. In this case, the function \(f(x)\) is given by the set \(\{(-16, 13), (-11, 82), (22, 48), (24, 26), (27, 31), (47, 98)\}\). Our objective is to extract these input values to determine the domain.
- Extracting the Domain Elements
To find the domain, we examine each ordered pair \((x, y)\) in the set and identify the \(x\)-component:
- For \((-16, 13)\), the first component is \(-16\).
- For \((-11, 82)\), the first component is \(-11\).
- For \((22, 48)\), the first component is \(22\).
- For \((24, 26)\), the first component is \(24\).
- For \((27, 31)\), the first component is \(27\).
- For \((47, 98)\), the first component is \(47\).
- Finalizing the Domain List
Gathering these first components into a set, we obtain the domain of the function. Listing them in ascending order for clarity, the domain is \(\{-16, -11, 22, 24, 27, 47\}\). Each value in this set is a valid input for the function, as each maps to exactly one output. Therefore, the domain of \(f(x)\) is \(\boxed{\{-16, -11, 22, 24, 27, 47\}}\).
Examples
In real-world applications, such as a database of product prices, the domain would represent the set of all unique product IDs. Each product ID (input) maps to exactly one current price (output). Identifying the domain allows a manager to quickly determine which items are currently available in the system. By listing all unique IDs, one can easily keep track of inventory and ensure no duplicates exist, maintaining data integrity in the system.