Find the domain of the function defined by the table below. Express your answer as a set of numbers.
$x$ | $y$ |
10 | -8 |
8 | -2 |
-3 | 6 |
The domain of a function is defined as the set of all possible input values, typically denoted by \(x\), for which the function provides an output value \(y\). When a function is given in the form of a table, the domain consists of all the values listed in the column designated for the input \(x\). Analyzing the provided table:
| \(x\) | \(y\) \) \(10\) | \(-8\) \) \(8\) | \(-2\) \) \(-3\) | \(6\) \) |
We identify the corresponding \(x\)-values as \(10\), \(8\), and \(-3\).
By collecting these distinct \(x\)-values into a set, we represent the domain mathematically. The set notation uses curly braces to group the elements. Thus, the set of all input values is \(\{10, 8, -3\}\). Conventionally, sets are often ordered from least to greatest, though the order does not change the set's composition. Rearranging them gives \(\{-3, 8, 10\}\).
The domain of the function, representing the collection of all valid input values identified from the table, is the set \(\{-3, 8, 10\}\). Thus, the domain is \(\boxed{\{-3, 8, 10\}}\).
In real-life scenarios, the domain of a function often corresponds to the "valid range of inputs" for a system. For instance, if you are designing a database for a library, the "ID" of each book serves as the input. If the library only contains books with specific IDs, those IDs form the domain of the search function. Identifying the domain helps ensure that any query made to the system with a value outside of this set results in an "invalid entry" error, keeping the system reliable and orderly.