Problem 33 - A Graphical Approach to College Algebra

Question

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\begin{array}{c|c|c|c|c|c} x & 0 & 1 & 2 & 3 & 4 \\ \hline y & \sqrt{2} & \sqrt{3} & \sqrt{5} & \sqrt{6} & \sqrt{7} \end{array}$$

Step-by-Step Solution

Answer

The domain is \( \{ 0, 1, 2, 3, 4 \} \), the range is \( \{ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7} \} \), and it is a function.

Step 1: Identify the Domain

The domain of a relation is the set of all possible input values (x-values). According to the given table, the domain \( D \) consists of the x-values provided: \( D = \{ 0, 1, 2, 3, 4 \} \).

Step 2: Identify the Range

The range of a relation is the set of all possible output values (y-values). According to the table, the range \( R \) includes the y-values: \( R = \{ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7} \} \).

Step 3: Determine if the Relation is a Function

A relation is a function if each input (x-value) is paired with exactly one output (y-value). In the table, each x-value is associated with exactly one y-value, so this relation is indeed a function.

Key Concepts

Domain

The concept of the **domain** is fundamental in understanding how relations and functions work in mathematics. Simply put, the domain refers to the complete set of possible input values for a given function or relation.
In our exercise, the domain is the set of x-values from the provided table. These values are the inputs that we consider for our function or relation.
The table clearly shows that the x-values are 0, 1, 2, 3, and 4. Therefore, for this specific relation, the domain is:

• \( D = \{ 0, 1, 2, 3, 4 \} \)

Identifying the domain is crucial because it defines the range of inputs that the function or relation can accept. When dealing with graphs, the domain often extends to its visual limits, but in tables, as in this example, the domain is only the inputs explicitly listed.

Range

Understanding the **range** of a function or relation is as important as understanding the domain. The range is all the possible output values (y-values) that a function or relation can produce from its domain.
Looking at the specific exercise, we need to establish what y-values our given x-values produce. From the table:

• x = 0 generates y = \( \sqrt{2} \)
• x = 1 generates y = \( \sqrt{3} \)
• x = 2 generates y = \( \sqrt{5} \)
• x = 3 generates y = \( \sqrt{6} \)
• x = 4 generates y = \( \sqrt{7} \)

Thus, the range is the collection of these outputs:

• \( R = \{ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7} \} \)

This forms the range of the relation, which is simply the set of resulting y-values. Knowing the range is important because it allows us to see the potential outcomes of the function or relation when we input the domain values.

Relations

A **relation** in mathematics refers to a set of ordered pairs, typically consisting of an input value from a domain and an output value from a range.
The exercise we are examining asks us to determine if a relation is a function. This relation, expressed as a table, has pairs like \((0, \sqrt{2})\) and \((1, \sqrt{3})\).
In this particular case, every x-value has a single unique y-value paired with it.
In mathematical terms, a relation is considered a **function** if:

• Each input from the domain maps to exactly one output in the range.

The given table satisfies this condition since:

• 0 is paired with \( \sqrt{2} \)
• 1 is paired with \( \sqrt{3} \)
• 2 is paired with \( \sqrt{5} \)
• 3 is paired with \( \sqrt{6} \)
• 4 is paired with \( \sqrt{7} \)

This pairing supports that the relation is indeed a function. Recognizing the nature of relations and understanding when they are functions is integral for advancing in mathematical studies, as it helps in predicting and modeling real-world scenarios.