Problem 61 - Algebra 2

Question

State the domain and range of each relation. Then graph the relation and determine whether it is a function. \(\\{(-1,5),(1,3),(2,-4),(4,3)\\}\)

Step-by-Step Solution

Answer

Domain: \{-1, 1, 2, 4\}; Range: \{5, 3, -4\}; It is a function.

Step 1: Identify the Domain

The domain of a relation is the set of all possible input values (x-values). For the given relation \( \{(-1,5),(1,3),(2,-4),(4,3)\} \), the x-values are -1, 1, 2, and 4. Therefore, the domain is \( \{-1, 1, 2, 4\} \).

Step 2: Identify the Range

The range of a relation is the set of all possible output values (y-values). For the given relation \( \{(-1,5),(1,3),(2,-4),(4,3)\} \), the y-values are 5, 3, -4, and again 3. Ignoring the repeat, the range is \( \{5, 3, -4\} \).

Step 3: Graph the Relation

Plot each pair on a coordinate plane: \((-1,5)\), \((1,3)\), \((2,-4)\), \((4,3)\). Connect the dots only if needed to visualize, keeping them as points to reflect the discrete nature.

Step 4: Determine if the Relation is a Function

To determine if a relation is a function, each x-value must pair with exactly one y-value. Check each pair: \(-1\) pairs with \(5\), \(1\) pairs with \(3\), \(2\) pairs with \(-4\), and \(4\) pairs with \(3\). Each x-value is paired with one unique y-value, so this relation is a function.

Key Concepts

Domain and Range

Understanding the domain and range is crucial in mathematics, especially in functions and relations. The domain of a relation refers to the set of all possible input values. These are commonly represented as the x-values in ordered pairs. So, if you have the relation \( \{(-1,5),(1,3),(2,-4),(4,3)\} \), the domain will be the set of x-values: \(-1, 1, 2,\) and \(4\).

On the other hand, the range of a relation consists of all possible output values, the y-values. For the same relation, these are \(5, 3, -4, \) and also \(3\) again. It's important to note that we disregard repeated values for the range. Hence, the range is \( \{5, 3, -4\} \).

Knowing the domain and range helps in understanding the behavior of relations on a coordinate plane.

Relations

A relation in mathematics can be thought of as a collection of ordered pairs. Each pair consists of a first element (the x-value) and a second element (the y-value). In our case, \((-1, 5)\) is an example of such a pair. Relations describe how elements from two sets maybe interconnected, without adding restrictions like functions do.

For instance, both \( (1,3)\) and \( (4,3)\) are part of the relation \(\{(-1,5),(1,3),(2,-4),(4,3)\}\). While they share a y-value, this doesn't impact the relation's status, since relations allow multiple x-values to link to the same y-value.

In the visual context, when you plot relations on a coordinate plane, you analyze the structure of data points derived from these ordered pairs.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graphically represent mathematical relations and functions. It's made up of two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical). Where these axes intersect is the origin, denoted by the coordinate \((0,0)\).

To graph the relation \( \{(-1,5),(1,3),(2,-4),(4,3)\} \) on the coordinate plane, each ordered pair is plotted as a point. So, the point \((-1,5)\) is plotted by moving 1 unit left on the x-axis and 5 units up on the y-axis. Continue this method for the others: \((1,3)\), \((2,-4)\), and \((4,3)\).

This visual representation aids in understanding the spatial distribution of the relation's domain and range.

Function Determination

Determining if a relation is a function is a fundamental concept in algebra. A relation is a function if each input (x-value) is associated with exactly one output (y-value). This means that in the entire set of ordered pairs, no x-value is repeated with a different y-value.

Taking our example relation \( \{(-1,5),(1,3),(2,-4),(4,3)\} \), observe each ordered pair. The x-values \(-1, 1, 2,\) and \(4\) each map to one specific y-value, namely \(5, 3, -4,\) and \(3\) respectively. Since no x-value is paired with more than one y-value, this relation satisfies the condition of being a function.

This evaluation process helps confirm the nature of mathematical relationships, ensuring consistency and predictable outcomes when dealing with functions.