Problem 27 - 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. $$ \\{(0,-1.1),(2,-3),(1.4,2),(-3.6,8)\\} $$

Step-by-Step Solution

Answer

The relation is a function with domain \([-3.6, 0, 1.4, 2]\) and range \([-3, -1.1, 2, 8]\); it is discrete.

Step 1: Plot the Points

To graph the relation, we need to plot each ordered pair on a coordinate plane. Use the pairs (0, -1.1), (2, -3), (1.4, 2), and (-3.6, 8) as your points. Mark them clearly on the graph so you can see their distribution.

Step 2: Determine the Domain

The domain of a relation is the set of all possible x-values (inputs). For the given relation, consider the x-coordinates: 0, 2, 1.4, and -3.6. Hence, the domain is: \([-3.6, 0, 1.4, 2]\).

Step 3: Determine the Range

The range of a relation is the set of all possible y-values (outputs). Look at the y-coordinates from the ordered pairs: -1.1, -3, 2, and 8. Thus, the range is: \([-3, -1.1, 2, 8]\).

Step 4: Check if it's a Function

A relation is a function if each x-value in the domain corresponds to exactly one y-value. Check the x-coordinates: 0, 2, 1.4, and -3.6 each map to a unique y-value. Therefore, the relation is a function.

Step 5: Determine if it's Discrete or Continuous

A discrete relation consists of distinct, separate points, while a continuous relation has no interruptions. Since the points are individual and not connected by a line, the relation is discrete.

Key Concepts

Understanding Domain and Range

The domain and range are fundamental concepts when exploring relations and functions. They help you understand the input and output values a relation can produce.

- **Domain**: This is the complete set of possible values of the independent variable, usually denoted by 'x', that a relation can have. In simple words, it's the list of all x-values you can use in the relation. For the given relation \((0,-1.1),(2,-3),(1.4,2),(-3.6,8)\), the domain comprises the x-values \([-3.6, 0, 1.4, 2]\).

- **Range**: Similarly, the range refers to all possible outputs or y-values of a relation. It encompasses the y-values obtained from employing each x-value in the domain. For this case, the range is \([-3, -1.1, 2, 8]\).

By analyzing domain and range, we gain a deeper insight into where a relation can operate and what values are obtainable.

Exploring Discrete and Continuous Relations

Relations can be classified as either discrete or continuous, which assists in understanding the nature and behavior of the relation you're dealing with.

- **Discrete Relations**: These are characterized by distinct, separate points without any connection. In other words, the x-values are isolated from each other, without forming a complete line or curve when plotted. The points in the example \((0,-1.1),(2,-3),(1.4,2),(-3.6,8)\) are not connected, demonstrating that the relation is discrete.

- **Continuous Relations**: In contrast, continuous relations create a smooth curve or line when plotted. The values smoothly transition from one to another without any gaps. These relations include all numbers in an interval.

For this example, observing the separate nature of the plotted points confirms that it is indeed a discrete relation.

Graphing Relations and Determining Functions

Graphing helps in visualizing the structure and behavior of a relation or function. **Graphing a relation** entails plotting each pair of x and y values on a coordinate plane. For our exercise, each ordered pair like \((0,-1.1)\) and \((2,-3)\) must be marked clearly on the graph.

After plotting, **determining if a relation is a function** is vital. A relation becomes a function if each x-value corresponds to exactly one y-value. An easy rule to test this is the "vertical line test". If a vertical line crosses the graph in more than one place at any point, it's not a function. In our example, each x-value \([-3.6, 0, 1.4, 2]\) links uniquely to a y-value \([-3, -1.1, 2, 8]\), signifying a function.

Understanding these categories is essential for deciphering how mathematical relations operate. With the graph, different aspects of the relation, such as continuity and functional nature, become readily apparent.