Problem 2 - Algebra 2

Question

Find the inverse of each relation. $$ \\{(1,3),(1,-1),(1,-3),(1,1)\\} $$

Step-by-Step Solution

Answer

The inverse is \( \{(3,1), (-1,1), (-3,1), (1,1)\} \).

Step 1: Understanding the Exercise

In this task, we are given a set of ordered pairs which forms a relation. We need to find the inverse of this relation by swapping the elements in each pair.

Step 2: Analyze the Given Relation

The relation is represented by the set of ordered pairs \( \{(1,3), (1,-1), (1,-3), (1,1)\} \). Each pair is in the form \((x, y)\), where \(x\) is the first element, and \(y\) is the second element.

Step 3: Swap Elements in Each Pair

To find the inverse of the relation, swap the elements of each pair so that the first element becomes the second, and the second element becomes the first. This means that the pair \((a, b)\) becomes \((b, a)\).

Step 4: Apply Swap Operation

Perform the swap on each pair from the original relation:\(\\begin{align*}&(1, 3) \to (3, 1) \&(1, -1) \to (-1, 1) \&(1, -3) \to (-3, 1) \&(1, 1) \to (1, 1)\end{align*}\\)

Step 5: Write Down the Inverse Relation

The inverse relation, after swapping each pair, is \( \{(3,1), (-1,1), (-3,1), (1,1)\} \).

Key Concepts

Ordered Pairs

Ordered pairs are fundamental elements in mathematics, especially when dealing with relations and functions. They are written as \((x, y)\), where the order of elements is crucial.

• The first element, usually denoted as \(x\), is known as the "first coordinate" or "input."
• The second element, denoted as \(y\), is called the "second coordinate" or "output."

Ordered pairs are used to represent positions on a grid or points in space. They are essential for plotting graphs and visualizing data. They provide a clear way to show a relationship between two quantities.
Understanding ordered pairs is vital when working with relations, as they depict how two factors interact. In exercises like finding an inverse relation, ordered pairs allow us to clearly see and manipulate the relationship between elements.

Relation

A relation in mathematics connects elements from one set to elements of another. It can be visualized as a collection of ordered pairs. For instance, the set \(\{(1, 3), (1, -1), (1, -3), (1, 1)\}\) is a relation.

• Here, each ordered pair is a mapping from one number to another, showing how they relate to each other.
• It’s possible for one element to be associated with multiple elements in the other set, as seen with our example where "1" is linked to several values.

Relations are more about mapping and connecting values rather than just matching them. Relations are useful for understanding how different quantities interact or connect in a mathematical sense. They form a basis for more advanced concepts such as functions and inverse relations.

Swap Operation

The swap operation is a simple yet powerful concept. It involves exchanging the positions of the elements in an ordered pair. For example, when given a pair \((a, b)\), swapping means reordering it as \((b, a)\).

• This operation is particularly important in finding the inverse of a relation.
• By swapping, you essentially reverse the roles of the input and output.

In our problem, swapping each pair in the given relation \((1, 3)\) to \((3, 1)\), and so on, transforms the relation to its inverse form.
This operation may seem straightforward, but it’s a key step in reversing relationships, making it an essential part of understanding inverse functions.

Inverse Function

An inverse function effectively "reverses" a given function or relation. It is formed by swapping the ordered pairs of a relation and rearranging their order.

• The inverse of a relation \(\{(1, 3), (1, -1), (1, -3), (1, 1)\}\) is achieved by swapping to get \(\{(3, 1), (-1, 1), (-3, 1), (1, 1)\}\).
• The process involves taking each \(x\) and \(y\) value and switching their positions.

When dealing with functions, it translates to finding a rule where the roles of inputs and outputs are reversed.This ensures that if the original function maps \(x\) to \(y\), the inverse function maps \(y\) back to \(x\).
Inverse functions are valuable for solving equations and real-world problems where cause and effect need to be undone or reversed.