Problem 3 - College Algebra

Question

determine whether each relation is a function. Give the domain and range for each relation. $$ [(3,4),(3,5),(4,4),(4,5)] $$

Step-by-Step Solution

Answer

The given relation is not a function. Its domain is \(\{3, 4\}\) and its range is \(\{4, 5\}\).

Step 1: Checking if the Relation is a Function

Look at the given relation: \[(3,4), (3,5), (4,4), (4,5)\]. An observation can be made that the number 3 in the domain (the 'x' value) corresponds to two different 'y' values in the range (4 and 5), similarly, the number 4 in the domain also corresponds to two different 'y' values (4 and 5). Since a particular 'x' value corresponds to more than one 'y' value, therefore this relation is not a function as per the definition.

Step 2: Determining the Domain

The domain of a function (relation) is the set of all 'x' values. So in this relation, the domain is the set of the first numbers from the ordered pairs. Therefore, Domain = \(\{3, 4\}\).

Step 3: Determining the Range

The range of a function (relation) is the set of all 'y' values. In this relation, the range is the set of the second numbers from the ordered pairs. Hence Range = \(\{4, 5\}\).

Key Concepts

Domain

The concept of the domain in mathematics refers to all the possible input values (usually denoted as 'x') that a function or a relation can accept. Essentially, it is the set of all first elements in each ordered pair within the relation. For example, consider the relation given as \([(3,4),(3,5),(4,4),(4,5)]\). By reviewing these pairs, we can easily identify the domain by collecting all the unique x-values:

• The first ordered pair is \((3,4)\), with the x-value of 3.
• The second ordered pair is \((3,5)\), also holding the x-value of 3.
• The third ordered pair is \((4,4)\), with an x-value of 4.
• The fourth ordered pair is \((4,5)\), also with an x-value of 4.

Therefore, combining all these unique x-values, the domain of this relation is \(\{3, 4\}\).
In any mathematical relation, understanding the domain helps to clarify the limits of the function's inputs. It is crucial because a function cannot have inputs outside of its domain.

Range

The range represents all the possible output values a function or a relation can produce. In terms of ordered pairs, it includes all the second elements. To find the range of a relation like \([(3,4),(3,5),(4,4),(4,5)]\), you would extract the y-values from each pair:

• From the pair \((3,4)\), the y-value is 4.
• From the pair \((3,5)\), the y-value is 5.
• From the pair \((4,4)\), the y-value is 4 again.
• From the pair \((4,5)\), the y-value is 5 again.

After gathering these y-values, we see that the formal range of this relation is \(\{4, 5\}\).
Determining the range is important as it shows all potential outputs of the function or relation. When you know the range, you gain insight into the behavior of the outputs based on different inputs in the domain.

Relations

Relations in mathematics express a connection between sets of values, typically described with ordered pairs. A relation consists of pairs where the first element represents the input (or domain), and the second element represents the output (or range).
For example, the relation \([(3,4),(3,5),(4,4),(4,5)]\) consists of ordered pairs where the x-values are related to y-values.
To determine if a given relation is a function, we check whether each x-value is associated with exactly one y-value:

• Here, the x-value of 3 is related to both y-values 4 and 5.
• Similarly, the x-value of 4 is related to y-values 4 and 5.

In this case, since each x-value connects to more than one y-value, this relation is not a function.
Understanding the distinction between relations and functions is key because functions have stricter rules, meaning every input must correspond to one and only one output. Recognizing whether a relation meets these criteria is fundamental to grasping more complex mathematical and real-world applications.