Problem 10 - Algebra 2

Question

Find the inverse of each relation. $$ \\{(2,6),(4,5),(-3,-1)\\} $$

Step-by-Step Solution

Answer

The inverse relation is \( \{(6,2),(5,4),(-1,-3)\} \).

Step 1: Understanding the Problem

We are given a set of ordered pairs \( \{(2,6), (4,5), (-3,-1)\} \). Our task is to find the inverse of this set of relations.

Step 2: Definition of Inverse Relation

The inverse of a relation swaps the first and second elements in each pair. If the original pair is \((a, b)\), the inverse will be \((b, a)\).

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a pair of elements written in a specific order. Typically, we use parentheses and a comma to separate the two elements, like this:

• First element: Often referred to as the "x-coordinate" in many contexts.
• Second element: Often referred to as the "y-coordinate."

Ordered pairs are fundamental in conveying relationships between variables. They define points in the coordinate system or elements within sets of relations. Because order matters, the pair \((2, 6)\) is distinct from \((6, 2)\). In the given set, \(\{(2,6), (4,5), (-3,-1)\}\), we consider each pair separately, maintaining the given order of elements. This is crucial when discussing inverses, as we'll explore next.

Functions

A function is a specific type of relation where each input is related to exactly one output. It is defined as a set of ordered pairs where no two distinct pairs have the same first element. This uniqueness of the first element makes functions predictable and reliable. Understanding functions this way allows us to easily visualize their behavior. Consider functions as machines that process inputs into outputs. For every input, there is a definite and singular output. For example, one possible function from the given set of ordered pairs could relate the x-coordinates to y-coordinates.

• If input is 2, output will be 6.
• If input is 4, output will be 5.
• If input is -3, output will be -1.

Functions form the basis for numerous mathematical concepts and daily applications, from calculating distances to computer algorithms.

Relation

In mathematics, a relation is a connection between a set of values. Relations consist of ordered pairs where the first element depends on or relates to the second element. These pairs are especially useful to show how different elements connect.For example, in the set \(\{(2,6), (4,5), (-3,-1)\}\), each pair gives us a clear connection between two numbers.

• The relation between 2 and 6,
• 4 and 5,
• and -3 and -1.

In terms of a mathematical relation, each pair states that there is a direct link from the first number to the second. In this context, to find the inverse relation, you simply swap the pairs to get \(\{(6,2), (5,4), (-1,-3)\}\). This inversion helps us understand how changes in one element reflect in another.

Domain and Range

Domain and range are essential concepts when dealing with relations and functions. The domain is the set of all first elements in the ordered pairs (inputs), while the range is the set of all second elements (outputs).Considering the original set \(\{(2,6), (4,5), (-3,-1)\}\), the domain is \(\{2, 4, -3\}\) and the range is \(\{6, 5, -1\}\).

• The domain includes every possible input, or every x-value in the set, indicating the start point for any relations or functions.
• The range includes corresponding outcomes for each input, showing the possible results or y-values.

When dealing with inverse relations, the domain and range effectively swap places. Therefore, the domain of the inverse relation \(\{(6,2), (5,4), (-1,-3)\}\) is \(\{6, 5, -1\}\) and the range becomes \(\{2, 4, -3\}\). This switch offers a deeper insight into how elements relate within inversions.