Problem 1 - Algebra 2

Question

Find the inverse of each relation. Graph the given relation and its inverse. $$ \begin{array}{|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} \\ \hline y & {0} & {1} & {0} & {2} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The inverse relation is {(0,1), (1,2), (0,3), (2,4)}. When both the given relation and its inverse are plotted on a graph, the graph of the inverse function appears as a reflection of the original function over the line y = x.

Step 1: Identify the Given Relation

The relation is given in terms of pairs of x and y values in a table. It can be represented as set of ordered pairs as {(1,0), (2,1), (3,0), (4,2)}.

Step 2: Find the Inverse Relation

To find the inverse of a relation, simply switch the x and y values. Therefore, the inverse relation is {(0,1), (1,2), (0,3), (2,4)}.

Step 3: Graph the Given Relation and Its Inverse

First plot the given relation points (1,0), (2,1), (3,0), (4,2) on a graph. Then plot the points of the inverse relation (0,1), (1,2), (0,3), (2,4) on the same graph. Notice that the graph of the inverse function is a reflection of the original function over the line y = x.

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a fundamental concept used to signify a pair of elements. These pairs are written in the form \((x, y)\), where order is crucial. This differs from sets where elements can be unordered. In the context of relations, ordered pairs depict a specific relationship between two values. For instance, in the given exercise, the ordered pairs are \((1,0), (2,1), (3,0), (4,2)\). Here, each element from the first column (x-values) matches an element from the second column (y-values). The relation is essentially a collection of these pairs.

Ordered pairs are essential in numerous mathematical applications, including functions, algebra, and geometry. They provide a clear way to specify the location of points in graphs, linking algebraic relationships with visual representations.

Graphing Relations

Graphing relations provides a visual means of understanding and analyzing mathematical relationships. When given a set of ordered pairs, graphing involves plotting these points on a coordinate plane. The x-value of each pair is plotted on the horizontal axis, while the y-value is plotted on the vertical axis.

In the given exercise, the ordered pairs \((1,0), (2,1), (3,0), (4,2)\) are graphed by placing a point at each corresponding coordinate. This visual representation helps identify patterns or trends within the data. For instance, points that form a straight line might suggest a linear relationship.

When graphing relations and their inverses, it's fascinating to observe how the inverse points \((0,1), (1,2), (0,3), (2,4)\) align in relation to the original set. It reveals the change when x-values and y-values are swapped, which is key in understanding inverse relations.

Reflection over y=x

The concept of reflection over the line \(y=x\) is central to understanding inverse relations graphically. This reflection is a transformation that swaps the x-coordinate and y-coordinate of each point in a relation.

In the exercise, when plotting both the original relation and its inverse, each point is reflected over the line \(y=x\). This symmetric line acts as a mirror, showing how points like \((1,0)\) become \((0,1)\) in the inverse plot.

• A point \((a, b)\) in the original relation will have its reflection at \((b, a)\) in the inverse.
• This reflection principle illustrates the swappable nature of the variables in functions and their inverses.
• Understanding this mirror-like quality helps in visualizing and verifying inverse relationships effortlessly.

Mastering this concept simplifies many algebraic problems, granting insight into the inherent symmetries of mathematical functions and relations.