Problem 31 - 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. $$\begin{array}{c|c|c|c|c} x & 11 & 12 & 13 & 14 \\ \hline y & -6 & -6 & -7 & -6 \end{array}$$

Step-by-Step Solution

Answer

Domain: \(\{11, 12, 13, 14\}\), Range: \(\{-6, -7\}\), and the relation is a function.

Step 1: Identify the Ordered Pairs

The table of values shows the relationship between \(x\) and \(y\). Each column represents an ordered pair. From the table, we have the ordered pairs: \((11, -6)\), \((12, -6)\), \((13, -7)\), and \((14, -6)\).

Step 2: Determine the Domain

The domain \(D\) of a relation is the set of all possible \(x\)-values. From our ordered pairs, the \(x\)-values are 11, 12, 13, and 14. Therefore, the domain is \(D = \{11, 12, 13, 14\}\).

Step 3: Determine the Range

The range \(R\) is the set of all possible \(y\)-values. The \(y\)-values from the ordered pairs are -6 and -7. Hence, the range is \(R = \{-6, -7\}\).

Step 4: Check if the Relation is a Function

To determine if the relation is a function, each \(x\)-value must correspond to exactly one \(y\)-value. In the ordered pairs \((11, -6)\), \((12, -6)\), \((13, -7)\), and \((14, -6)\), each distinct \(x\)-value corresponds to only one \(y\)-value. Therefore, the relation is a function.

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics used to show a relationship between two elements. Each ordered pair consists of two elements: the first element, usually denoted as \(x\), and the second element, signified as \(y\). These pairs are noted as \((x, y)\). In the context of the given exercise, the columns in the table represent ordered pairs. Understanding ordered pairs is crucial because it forms the basis for more complex concepts like functions and relations. You'll often find them in coordinate systems where they serve to pinpoint locations on a graph. Each \(x\) value in the ordered pair is a point on the horizontal axis, while the \(y\) value corresponds to a point on the vertical axis.Here are some characteristics of ordered pairs:

• The order in which the numbers are written is crucial. \((11, -6)\) is not the same as \((-6, 11)\).
• Ordered pairs can represent solutions to equations or can be used in defining relations and functions (more about that below).

By recognizing these ordered pairs in the table, you can easily determine the domain and range of a relation, which are sets formed from \(x\) and \(y\) values, respectively.

Function

A function is a specific type of relation where each element in the domain is related to exactly one element in the range. In simpler terms, in a function, no two ordered pairs have the same first element (\(x\)) with different second elements (\(y\)). This ensures that every input has a unique output.To check if a relation is a function, look at the list of ordered pairs. If you have any repeated \(x\)-values associated with different \(y\)-values, then it's not a function. In our exercise, we examined a set of ordered pairs:

• \((11, -6)\)
• \((12, -6)\)
• \((13, -7)\)
• \((14, -6)\)

Notice how each \(x\)-value (11, 12, 13, and 14) maps to a single \(y\)-value (-6 or -7). Thus, the relation qualifies as a function since no \(x\)-value is linked with multiple \(y\)-values.Understanding functions is beneficial since they are central to various fields like algebra, calculus, and computer science. Recognizing whether a relation is a function helps simplify analysis and problem-solving across different domains.

Relations

Relations in mathematics are a broader concept that encompasses functions. A relation is a set of ordered pairs, usually denoted as \((x, y)\). For any relation, you can derive the domain, which is the set of all \(x\)-values, and the range, which is the set of all \(y\)-values.In our exercise, the relation is represented by a table that lists specific \((x, y)\) pairs:

• \((11, -6)\)
• \((12, -6)\)
• \((13, -7)\)
• \((14, -6)\)

From these ordered pairs, we can determine:

• The domain, \(D = \{11, 12, 13, 14\}\), which includes all the \(x\)-values.
• The range, \(R = \{-6, -7\}\), which includes all the \(y\)-values.

While every function is a relation, not all relations are functions. A relation can have an \(x\)-value connected to more than one \(y\)-value and still be a valid relation. However, recall that in a function, every \(x\) must map to only one \(y\).Understanding relations allows us to understand how different mathematical entities are interconnected, which provides a foundation for exploring more advanced topics like functions and mappings.