Problem 5 - College Algebra

Question

determine whether each relation is a function. Give the domain and range for each relation. $$ \\{(3,-2),(5,-2),(7,1),(4,9)] $$

Step-by-Step Solution

Answer

Yes, the relation is a function. The domain is {3, 5, 7, 4} and the range is {-2, 1, 9}.

Step 1: Checking if the relation is a function

In order to check if the relation is a function, it is necessary to ensure each input (first element in each pair) maps to exactly one output (second element in each pair). Looking at the set of the ordered pairs, it can be seen that every input is only matched with one unique output. Hence, the relation is a function.

Step 2: Determining the domain

The domain is the set of all first elements from the given pairs. Extracting the first elements from each pair, the domain for this set is \{3, 5, 7, 4\}

Step 3: Determining the range

The range is the set of all second elements from the given pairs. Extracting the second elements from each pair, the range for this set is: \{-2, 1, 9\}

Key Concepts

Relations

In mathematics, a relation refers to a set of ordered pairs, where each pair consists of two elements. These elements are known as the input and output, or simply the first and second elements of the pair. A relation shows how the input and output are connected. The connection between these pairs can be represented in many ways, such as a table, graph, or list of pairs.

It is important to check whether a relation is a function. This is because not all relations are functions. A function is a special kind of relation where each input is paired with exactly one output. This means no input repeats with a different output.

For example, consider the relation \( \{(3,-2),(5,-2),(7,1),(4,9)\} \). Here, every input (3, 5, 7, and 4) is linked with one unique output (-2, -2, 1, and 9 respectively). Thus, this is a function because no input is paired with different outputs.

Domain

The domain of a relation or function is the complete set of possible input values. In simple terms, it is a collection of all the unique first elements from each ordered pair in the relation. Identifying the domain helps understand the inputs that were used in the relation.

To determine the domain of a set of ordered pairs, list out each unique first element. Using the earlier example, the ordered pairs are \((3,-2)\), \((5,-2)\), \((7,1)\), and \((4,9)\).

• The first elements, or inputs, are: 3, 5, 7, and 4.

Therefore, the domain of this relation is \{3, 5, 7, 4\}. Each number in this set represents an input that has a corresponding output.

Range

The range of a relation refers to the set of all possible outputs or second elements from the ordered pairs. It tells us the values that are achieved by using the domain. The range is like a reflection of the domain's counterparts in a given relation.

To find the range, you need to extract all the second elements from each pair and ensure they are unique. In the example \((3,-2)\), \((5,-2)\), \((7,1)\), and \((4,9)\),

• The second elements, or outputs, are: -2, -2, 1, and 9.

However, we only list unique values for the range, which results in the set \{-2, 1, 9\}. This set indicates all the outputs that the relation covers.