Problem 14 - Algebra 2

Question

Find the inverse of each relation. $$ \\{(6,11),(-2,7),(0,3),(-5,3)\\} $$

Step-by-Step Solution

Answer

The inverse relation is \(\{(11,6), (7,-2), (3,0), (3,-5)\}\).

Step 1: Understanding the Problem

To find the inverse of a relation, we need to swap the elements in each ordered pair. Essentially, if we have a pair \((a, b)\), its inverse will be \((b, a)\). The relation provided is a set of ordered pairs, and we'll apply this principle to each pair.

Step 2: Inverting Each Pair

We have the relation \(\{(6,11),(-2,7),(0,3),(-5,3)\}\). For each pair, swap the first and second elements: 1. \((6,11)\) becomes \((11,6)\).2. \((-2,7)\) becomes \((7,-2)\).3. \((0,3)\) becomes \((3,0)\).4. \((-5,3)\) becomes \((3,-5)\).

Step 3: Forming the Inverse Relation

The pairs from the previous step form the inverse relation. So, the inverse of the given relation is \(\{(11,6), (7,-2), (3,0), (3,-5)\}\). Check that each pair is correctly transformed by ensuring each second coordinate in the original pair becomes the first in the inverse.

Key Concepts

Ordered Pairs

Ordered pairs are fundamental elements in mathematics, especially when dealing with relations and functions. An ordered pair is written as \((x, y)\), representing a point or a pair of values. The first element, often called the *x*-coordinate, determines the horizontal position; the second element, the *y*-coordinate, determines the vertical position.
In essence, ordered pairs are used to specify locations on the coordinate plane.
One critical aspect of ordered pairs is that the sequence in which the numbers appear matters.

• For instance, \((1, 2)\) is different from \((2, 1)\), as they indicate different points in space or different elements involved in a relation.
• Ordered pairs help define relations, which are simply sets of such pairs. A relation maps elements from one set to another.
• The inverse of a relation flips these ordered pairs such that every \((a, b)\) becomes \((b, a)\).

Understanding ordered pairs is crucial to grasping the concept of inverse relations, providing a deeper insight into how different elements associate with each other.

Coordinate System

The coordinate system, also known as a Cartesian coordinate system, is a framework that allows us to identify the position of points in space using numbers. It typically consists of two perpendicular lines known as the *x-axis* and the *y-axis*.
This system is essential in visualizing relations and functions, represented graphically by plotting ordered pairs.
Each ordered pair \((x, y)\) corresponds to a unique point on this plane.

• The *x-coordinate* tells us how far along the horizontal axis the point is.
• The *y-coordinate* shows the point's position on the vertical axis.

In the given exercise, the set of ordered pairs is essentially plotting points on this coordinate system. Another important feature of this system is its capacity to transform these points by finding inverses, which involves switching the *x* and *y* coordinates of each pair, changing their graphical position on the plane. Thus, a solid understanding of the coordinate system is foundational for interpreting and manipulating mathematical relations effectively.

Algebra 2

Algebra 2 is a stepping stone in high school mathematics, focusing on a broad range of concepts including functions, systems of equations, and problem-solving techniques.
One area of study involves understanding relations and inverses, as highlighted in the current exercise. In Algebra 2, students learn to interpret and find the inverse of various relations, providing a contrasting perspective to ordinary functions.
In the context of inverses,

• Swapping coordinates of ordered pairs is a primary method to find the inverse of a given relation.
• Understanding how to rearrange these elements not only helps in mathematical graphing but also extends into real-world applications such as computer graphics and data mapping.

Algebra 2 reinforces a student's mathematical toolkit, ensuring they can tackle complex problems by understanding these fundamental concepts. This allows for advanced study in fields that rely heavily on data visualization and algebraic manipulation.