- The domain of a function consists of all input (x) values, while the range consists of all output (y) values.
- Identify the y-values from the table: \(\{9, 0, -7, -1\}\).
- Sort the y-values in ascending order: \(\{-7, -1, 0, 9\}\).
- The correct range is \(\boxed{\{y \mid y=-7,-1,0,9\}}\).
Explanation
- Understanding the Definition of Range
The problem asks for the range of a function defined by a table. The domain and range of a function are fundamental concepts in algebra. The domain is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values) that the function produces. We are given the following mapping:
\[ \begin{array}{|l|l|} \ x & y \\ -5 & 9 \\ 1 & 0 \\ 4 & -7 \\ 6 & -1 \\ \end{array} \]
- Extracting and Ordering the Y-values
To identify the range, we need to extract all the unique output values (y-values) from the second column of the table. The values listed under the y-column are 9, 0, -7, and -1. Collecting these into a set, we have \(\{9, 0, -7, -1\}\). To present this set clearly and according to standard mathematical convention, we order the elements from the smallest to the largest: \(\{-7, -1, 0, 9\}\).
- Selecting the Correct Option
We now compare our set \(\{y \mid y=-7, -1, 0, 9\}\) to the given options:
- \(\{x \mid x=-5, 1, 4, 6\}\): This represents the domain, not the range.
- \(\{y \mid y=-7, -1, 0, 9\}\): This matches our set of y-values perfectly.
- \(\{x \mid x=-7, -5, -1, 0, 1, 4, 6, 9\}\): This is an incorrect set.
- \(\{y \mid y=-7, -5, -1, 0, 1, 4, 6, 9\}\): This is also an incorrect set.
Therefore, the correct range of the given function is \(\{y \mid y=-7, -1, 0, 9\}\).
Examples
Understanding the range of a function is crucial in real-world scenarios, such as modeling temperature changes. If a sensor records temperatures at different times, the input values (x) represent the time, and the range (y) represents all the unique temperatures recorded. Identifying the range helps us determine the minimum and maximum extremes of a process, which is essential for climate control, engineering safety, and data analysis.