Problem 25 - Algebra 2

Question

Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function and state whether it is discrete or continuous. $$ \\{(-2,5),(3,7),(-2,8)\\} $$

Step-by-Step Solution

Answer

Domain: \(-2, 3\); Range: \(5, 7, 8\); Not a function; Discrete.

Step 1: Plot the Points

The given relation consists of three points: (-2,5), (3,7), and (-2,8). Plot each of these points on a Cartesian coordinate graph, where the x-axis represents the first element of each pair and the y-axis represents the second element.

Step 2: Analyze the Domain

The domain of a relation is the set of all x-values from the ordered pairs. Here, identify the unique x-values: -2 and 3. Thus, the domain is \(-2, 3\).

Step 3: Analyze the Range

The range of a relation is the set of all y-values from the ordered pairs. Extract and list the unique y-values: 5, 7, and 8. Therefore, the range is \(5, 7, 8\).

Step 4: Determine If It Is a Function

A relation is a function if each x-value is paired with exactly one y-value. Check the given points: the x-value -2 is paired with both 5 and 8, indicating repetition with different y-values. Hence, this relation is not a function.

Step 5: Identify If Discrete or Continuous

Since the relation consists of separate points rather than a connected line or curve, it is discrete. Continuous relations do not have gaps, whereas discrete ones do, like in this case.

Key Concepts

Domain and Range

In the context of relations and functions in mathematics, understanding the concepts of domain and range is essential. The domain of a relation refers to the complete set of possible input values, typically noted as the set of x-values in ordered pairs. These are the values that feed into our function or relation. For the relation \{(-2,5),(3,7),(-2,8)\}, to find the domain, we identify all distinct x-values provided by the ordered pairs: -2 and 3. Hence, the domain in this case is \(-2, 3\).

The range, meanwhile, is about what values come out of the relation — it is the set of all possible output values. By examining the peculiar y-values in the ordered pairs, we get the set \(5, 7, 8\). So, our range for this relation would be \(5, 7, 8\). Understanding these two components is crucial in graphing the relation and in determining its wider properties.

Discrete and Continuous

When categorizing relations, one useful concept is whether they are discrete or continuous. A relation is considered discrete if it consists of distinct or separate points. These relations are not smooth or unbroken and often appear as isolated dots on a graph. In our exercise, with points \((-2, 5), (3, 7), (-2, 8)\), each point stands alone with no connecting lines. This scattered formation makes the relation discrete.

On the other hand, continuous relations show an unbroken line or curve, indicating that every x-value within an interval has a corresponding y-value. An example of a continuous function could be the graph of a parabola or a line. Such relations are smooth, without jumps or gaps. Recognizing whether a relation is discrete or continuous helps in predicting its behavior and in choosing suitable methods for analysis and display.

Graphing Relations

Plotting the graph of a relation involves transferring points, defined by ordered pairs, onto a coordinate plane. This visual representation aids in understanding the nature and properties of the relation. In our example, we graph the points \((-2, 5), (3, 7), (-2, 8)\) on a cartesian plane. Start by locating these points based on their x and y coordinates — the first number in each pair represents the x-coordinate, and the second number represents the y-coordinate.

Once plotted, these points provide a clearer picture of the relation. For our relation, plotting reveals that there are duplicate x-values, specifically \(-2\), paired with different y-values. This non-unique pairing indicates that the relation is not a function, since functions require that each input corresponds to one and only one output.

The graph allows you to quickly determine critical characteristics such as whether the relation is a function by inspecting how many y-values are incident upon any x-value, as well as discern whether the points form a discrete or continuous structure. Thus, graphing is an invaluable tool in analyzing and interpreting relations and functions.