Problem 33 - Elementary and Intermediate Algebra

Question

Determine whether the relation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x\). See Example 2. $$ \begin{array}{|r|r|} \hline x & y \\ \hline-4 & 6 \\ -1 & 0 \\ 0 & -3 \\ 2 & 4 \\ -1 & 2 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The relation is not a function as \(x = -1\) corresponds to \(y = 0\) and \(y = 2\). Thus, it fails the definition of a function.

Step 1: Understand the Definition of a Function

A relation from a set of inputs (domain) to a set of possible outputs (range), where each input is related to exactly one output, defines a function. This means for every value of \(x\), there should be only one corresponding value of \(y\).

Step 2: Analyze the Given Table

Examine the values of \(x\) in the table: \(-4, -1, 0, 2, -1\). Notice that \(-1\) appears twice, associated with different \(y\) values (0 and 2). This indicates a potential issue with the definition of a function.

Step 3: Identify if the Relation is a Function

Check if there are any \(x\) values associated with more than one \(y\) value. Since \(x = -1\) is paired with both \(y = 0\) and \(y = 2\), this shows that the relation is not a function. A valid function would assign a single \(y\) value to each \(x\) value.

Step 4: Provide the Required Ordered Pairs

The ordered pairs where a single \(x\) value is related to multiple \(y\) values are \((-1, 0)\) and \((-1, 2)\). This confirms that the given relation does not satisfy the criteria for being a function of \(x\).

Key Concepts

Relations

In mathematics, a relation describes a connection or link between elements of two sets. Relations are used to express how values from one set, often called the domain, are associated with values from another set, known as the range. Relations can be represented in various forms, including sets of ordered pairs, tables, diagrams, or equations.

Understanding relations is crucial as they provide a foundation for exploring more complex concepts like functions. A relation does not necessarily have to be a function; a function is a special type of relation with additional restrictions. In this context, it's vital to remember that relations can be many-to-many, one-to-many, many-to-one, or one-to-one. For a given exercise, it's important to discern what type of relation is presented before concluding if it is a function.

Ordered Pairs

Ordered pairs form the basic building blocks for defining relations. An ordered pair consists of two elements: the first element is from the domain and is often depicted as "x," while the second, usually represented as "y," belongs to the range. In notation, an ordered pair is written as \( (x, y) \).

The essence of ordered pairs is the order; \( (x, y) \) is distinct from \( (y, x) \), unless \( x = y \). This structure enables clear mappings from inputs to outputs. For functions, each "x" in an ordered pair must map strictly to one "y." If a single "x" value relates to multiple "y" values in ordered pairs, as shown in the given exercise \( (-1, 0) \) and \( (-1, 2) \), this signals that the relation is not a function.

Hence, ordered pairs are essential in understanding if a relation meets the criteria of a function or remains a more generalized relation. They offer precise insights into how elements from different sets interact with one another.

Domain and Range

The concepts of domain and range are pivotal in understanding relations and functions. The domain of a relation or function is the set of all possible inputs, which are the "x" values. In contrast, the range is composed of all possible outputs, which are the resulting "y" values.

For the given exercise, the domain is \( \{-4, -1, 0, 2\} \), while the range includes \( \{6, 0, -3, 4, 2\} \). When assessing whether a relation qualifies as a function, one should ensure that each value in the domain maps to precisely one outcome in the range. If any domain value maps to more than one range value, the relation fails to be a function, as evidenced by the repeated "x" in the ordered pairs \ \((-1,0) \) and \ \((-1,2) \).

Understanding domain and range aids in analyzing and interpreting the nature of relations, enabling us to categorize and identify them effectively as functions or non-functions.