- Extract the set of ordered pairs for each table.
- Define inverse functions as the set where inputs and outputs are swapped: if \((a, b)\) is in table A, \((b, a)\) must be in table B.
- Observe that the set of ordered pairs in Table 1 is \(\{(-5, 4), (-3, 0), (0, -6), (2, -10), (4, -14)\}\), which swaps to \(\{(4, -5), (0, -3), (-6, 0), (-10, 2), (-14, 4)\}\).
- Notice this matches the set of pairs in Table 4, identifying them as inverse functions.
- The correct options are \(\boxed{1, 4}\).
Explanation
- Understanding the Inverse Relationship
We are given five tables of values, each representing a function, and we need to identify which two of these tables represent inverse functions. Two functions \(f(x)\) and \(g(x)\) are inverses if, for every ordered pair \((a, b)\) in function \(f\), the corresponding pair \((b, a)\) is present in function \(g\). Our goal is to extract the ordered pairs for each table and test for this symmetry property.
- Comparing the Tables
First, we represent each table as a set of ordered pairs:
- Table 1: \(S_1 = \{(-5, 4), (-3, 0), (0, -6), (2, -10), (4, -14)\}\)
- Table 2: \(S_2 = \{(-14, -4), (-10, -2), (-6, 0), (0, 3), (4, 5)\}\)
- Table 3: \(S_3 = \{(-5, -14), (-3, -10), (0, -6), (3, 0), (9, 4)\}\)
- Table 4: \(S_4 = \{(-14, 4), (-10, 2), (-6, 0), (0, -3), (4, -5)\}\)
- Table 5: \(S_5 = \{(-5, -4), (-3, 0), (0, 6), (2, 10), (4, 14)\}\)
We compare the sets to find one where the inputs and outputs are swapped. Let's analyze Table 1 and Table 4. For Table 1, the points are \((-5, 4), (-3, 0), (0, -6), (2, -10), (4, -14)\). If we swap the coordinates of these points, we get: \((4, -5), (0, -3), (-6, 0), (-10, 2), (-14, 4)\). Comparing these with the set \(S_4 = \{(-14, 4), (-10, 2), (-6, 0), (0, -3), (4, -5)\}\), we see that all swapped points from Table 1 exist in Table 4.
- Conclusion
The comparison shows that Table 1 and Table 4 are indeed inverse functions because every point \((x, y)\) in Table 1 corresponds to a point \((y, x)\) in Table 4. Therefore, Table 1 and Table 4 are the correct pair of tables. The final answer is the choice 1 and 4.
Examples
Inverse functions are essential in fields like finance and engineering. For example, if a function calculates the amount of interest earned on an investment after a certain number of years, its inverse function can determine how many years it would take for an investment to reach a specific interest amount. Being able to identify inverse tables helps in reversing data-driven processes efficiently without re-computing the entire function from scratch.