- The inverse of a relation \(f(x)\) is obtained by swapping the \(x\) and \(y\) coordinates of each ordered pair \((x, y)\) in the set.
- For the given relation \(f(x) = \{(8, 3), (4, 1), (0, -1), (-4, -3)\}\), swap the values in each pair: \((8, 3) \to (3, 8)\), \((4, 1) \to (1, 4)\), \((0, -1) \to (-1, 0)\), and \((-4, -3) \to (-3, -4)\).
- The resulting inverse relation is \(g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\}\).
- The correct option is \(\boxed{\{(3, 8), (1, 4), (-1, 0), (-3, -4)\}}\).
Explanation
- Understanding the Problem
The given relation \(f(x)\) is defined as a set of ordered pairs: \(f(x) = \{(8, 3), (4, 1), (0, -1), (-4, -3)\}\). Each ordered pair consists of an input \(x\) and its corresponding output \(y\), represented as \((x, y)\). Our objective is to determine the inverse relation \(g(x)\) of the given function \(f(x)\).
- Defining the Inverse Relation
The inverse of a function or relation, denoted by \(g(x)\), is found by interchanging the coordinates of each ordered pair in the original relation. Specifically, if a point \((a, b)\) belongs to the relation \(f(x)\), then the point \((b, a)\) must belong to the inverse relation \(g(x)\). This means for every ordered pair \((x, y)\) in \(f(x)\), the inverse relation will contain the corresponding pair \((y, x)\).
- Determining the Inverse Relation Pairs
Now, we apply this rule to each pair in \(f(x)\):
- The pair \((8, 3)\) becomes \((3, 8)\).
- The pair \((4, 1)\) becomes \((1, 4)\).
- The pair \((0, -1)\) becomes \((-1, 0)\).
- The pair \((-4, -3)\) becomes \((-3, -4)\).
Combining these transformed pairs, the inverse relation is \(g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\}\).
- Conclusion
Comparing our calculated set \(g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\}\) with the given options, we identify that the last option matches our result. Therefore, the correct inverse relation is \(\{(3, 8), (1, 4), (-1, 0), (-3, -4)\}\).
Examples
In real-world applications, identifying an inverse relation is equivalent to reversing the process of an input-output system. For instance, if a temperature conversion formula relates Celsius to Fahrenheit as \(f(C) = F\), the inverse \(f^{-1}(F) = C\) helps convert temperature values back from Fahrenheit to Celsius. Similarly, in logistics or inventory tracking, if you have a database mapping item IDs to shelf locations, the inverse relation allows you to look up which item is stored at a specific location, demonstrating the practical importance of understanding inversions.