Question

Find the domain and range of each relation. Then determine whether the relation represents a function. \\{(-4,4),(-3,3),(-2,2),(-1,1),(-4,0)\\}

Short answer

Domain: \(-4, -3, -2, -1\). Range: \(4, 3, 2, 1, 0\). Not a function.

Step-by-step solution

- Identify all the Input (x-values)

List all the x-values from each of the ordered pairs: -4, -3, -2, -1, -4.

- Determine Unique Input Values for Domain

The domain consists of all unique x-values. In this case, the domain is \(-4, -3, -2, -1\). The x-value -4 appears twice, but only counts once in the domain.

- Identify all the Output (y-values)

List all the y-values from each of the ordered pairs: 4, 3, 2, 1, 0.

- Define the Range

The range consists of all unique y-values. In this case, the range is \(4, 3, 2, 1, 0\).

- Check if the Relation is a Function

A relation is a function if each input (x-value) maps to exactly one output (y-value). In this case, the input -4 maps to both 4 and 0, so the relation is not a function.

Key concepts

input values

The first step in analyzing a relation involves identifying all the input values. These are the x-values in the ordered pairs given. For example, in the relation \(\{(-4,4),(-3,3),(-2,2),(-1,1),(-4,0)\}\), the input values are -4, -3, -2, -1, and -4. Notice that some x-values may repeat.
When listing the input values, make sure to include every x-value, even if it appears more than once. Identifying the input values correctly is crucial for further steps, such as determining the domain and checking if a relation is a function.
This step lays the foundation for a deeper understanding of how the relation works and how the values interact. Understanding input values helps in making sense of the domain and simplifies the overall grasp of mathematical relations.

unique values

Once all the input values have been listed, the next step is to determine the unique values among them. These unique values form what is known as the 'domain' of the relation. In our example relation, after listing the input values as \(-4, -3, -2, -1, -4\), we see that -4 appears twice. However, the domain only includes each number once. Hence, the domain of this relation is \(-4, -3, -2, -1\).
Similarly, the output values (y-values) need to be considered for determining the range. In this case, the output values are 4, 3, 2, 1, 0, listed from the ordered pairs.
Ensuring that only unique values are included in the domain and range is important because repetition can lead to incorrect conclusions. Unique values simplify the portrayal of relations and functions, making it easier to understand and analyze them.

function determination

Determining whether a relation represents a function is an integral part of understanding mathematical relations. A relation is considered a function if each input (x-value) maps to exactly one output (y-value). This means that no x-value should be associated with more than one y-value.
In our example, the relation \(\{(-4,4),(-3,3),(-2,2),(-1,1),(-4,0)\}\) was found to have an x-value (-4) mapping to both 4 and 0. Therefore, it violates the condition needed for a relation to be a function. In simpler terms, some x-values are paired with multiple y-values, disqualifying it as a function.
Understanding this concept ensures clarity when analyzing relations and helps in identifying true functions. This knowledge is fundamental when progressing to more advanced topics in mathematics where functions play a crucial role.