Problem 14 - Precalculus

Question

Graph the given relation. $$ \\{(x,-3) \mid-4

Step-by-Step Solution

Answer

A horizontal line at \( y = -3 \) from just above \( x = -4 \) to \( x = 4 \).

Step 1: Understanding the Relation

The given relation \( \{(x, -3) \mid -4 < x \leq 4 \} \) can be interpreted as a set of points on a graph where the y-coordinate is fixed at -3, and the x-coordinate varies between -4 and 4. That means \( x \) can take any value greater than -4 and up to 4.

Step 2: Consider the Domain

The domain of the relation is expressed as \( -4 < x \leq 4 \), meaning that \( x \) starts just above -4 and ends at 4 (inclusive). In graph terms, this is a continuous interval without including the endpoint -4 but including 4.

Step 3: Identify the Range

For this relation, the range is very simple: the y-coordinate is always -3. Therefore, the range is the single value \( -3 \), regardless of the value of \( x \).

Step 4: Plotting the Points

To graph this relation, mark a horizontal line on the coordinate plane at \( y = -3 \). This line should start just above \( x = -4 \) and run continuously to \( x = 4 \). Use an open circle at \( x = -4 \) to show that -4 is not included, and a closed circle at \( x = 4 \) to indicate that 4 is included in the graph.

Step 5: Review the Graph

Examine the graph to ensure it reflects the relation: a horizontal line at \( y = -3 \) from just beyond \( x = -4 \) to \( x = 4 \). This ensures that all points on the line adhere to the relation's conditions.

Key Concepts

Domain and Range

When we're graphing a relation, two important concepts to understand are the domain and the range. The domain refers to all the possible values that the input (or x-values) can take. In this example, the domain is given by the inequality \(-4 < x \leq 4\). This means the x-values can start just after -4 and can go up to and include 4. Remember, an open circle indicates that the exact value (here -4) is not part of the domain, while a closed circle means the value (here 4) is part of the domain. This can be visualized on a number line or graph where there is an open circle at x = -4 and a closed circle at x = 4.
The range, on the other hand, is quite straightforward in this specific relation. Since all the points have a fixed y-value of -3, the range is just this single value: -3. It doesn't vary with changes in x, simplifying our graph analysis considerably.

• Domain: Set of x values between -4 and 4, not including -4 but including 4.
• Range: The constant y-value of -3 for all x in the domain.

Being familiar with how to identify domain and range helps clarify what part of the graph should be included, and which values the math equation takes into account.

Inequalities

Inequalities play a central role in expressing restrictions within a given set of numbers or values. In mathematics, they are used to describe a range of possible values in a concise manner.For instance, the inequality \(-4 < x \leq 4\) tells us a lot about the potential x-values. - The symbol "<" indicates that the value is strictly less, meaning that x cannot actually be -4. Instead, it can only approach it but remain greater than it.- Conversely, "\(\leq\)" indicates that x can be equal to 4 and all values are as large as 4.Understanding these symbols helps you recognize how the domain is bounded and whether endpoints are included or excluded. Visualizing these endpoints on a graph with open or closed circles helps to reinforce this concept, making it clear which values of x are possible and which aren't.Bullet points summarizing inequalities:

• "<": Excludes the value, represented by an open circle.
• "\(\leq\)": Includes the value, represented by a closed circle.

By mastering inequalities, you will gain an essential tool for graphing and solving math problems, rendering it easier to identify allowed values in a given context.

Horizontal Lines

Horizontal lines are a distinct feature in graphing relations. They occur when all y-coordinates associated with given x-values are the same. In the problem we evaluated, the line is at \( y = -3 \), showing that no matter what the x-value is, the y remains consistent.Key characteristics of horizontal lines:

• They have a slope of zero. This is because the rise (change in y) is zero as we move along the line.
• Visually, they run parallel to the x-axis.

These lines indicate a simple relationship between the variables—mostly a lack of relationship for changing x-values. Each point on a horizontal line can be described as (x, -3) as in our scenario, where x can vary, but y is always -3.Horizontal lines effectively show fixed y-values despite varying x-values, making them both easy to recognize and easy to graph. Recognizing and plotting horizontal lines helps us immediately understand that the values of y do not change, providing a stable visual representation of certain mathematical conditions.