Problem 32 - 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. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 30 & 2 \\ 30 & 4 \\ 30 & 6 \\ 30 & 8 \\ 30 & 10 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The relation is not a function. Examples: (30, 2) and (30, 4).

Step 1: Understanding the Definition of a Function

A relation is a function if each input in the domain corresponds to exactly one output in the range. In simpler terms, each value of \(x\) should map to only one value of \(y\).

Step 2: Analyzing the Given Relation

The given table shows multiple \(y\) values (2, 4, 6, 8, and 10) corresponding to the same \(x\) value, which is 30.

Step 3: Identifying a Violation of a Function

Since one \(x\) value (30) corresponds to multiple \(y\) values in the table, this violates the definition of a function.

Step 4: Citing Examples of Ordered Pairs with More Than One y Value per x Value

An example of such pairs from the table are (30, 2) and (30, 4). These are both ordered pairs with the same \(x\) value but different \(y\) values.

Key Concepts

Relation Definitions

In algebra, a **relation** is a simple connection between sets of values. This connection is often depicted as pairs of numbers. Usually, we have two sets: the domain and the range. The domain is all possible inputs, often denoted by the variable \(x\), and the range is all possible outputs, typically denoted by the variable \(y\).
For any relation, we pair each item from the domain with at least one item from the range. However, it is essential to distinguish between a general relation and a specific type of relation known as a function. For a relation to be classified as a function, each element from the domain should be paired with exactly one element from the range. This means that if a relation pairs one \(x\) value with more than one \(y\) value, it is **not** a function.
Understanding these definitions helps in identifying whether a given relation meets the criteria of a function or not.

Ordered Pairs in Algebra

**Ordered pairs** play a crucial role in understanding relations in algebra. An ordered pair is represented as \((x, y)\) and signifies a relationship between two elements where \(x\) is an input from the domain and \(y\) is an output from the range.
In an ordered pair, the order of \(x\) and \(y\) is very important: \((x, y)\) is different from \((y, x)\). For example, in the given table in the exercise, pairs like \((30, 2)\) and \((30, 4)\) exemplify how different \(y\) values connect to the same \(x\) value.
In the analysis of relations, examining such pairs provides insight into whether or not a collection of points is a function. If we observe multiple \(y\) values for the same \(x\) value, as shown in the exercise, it becomes evident that we are dealing with a basic relation, not a function.

Function Criteria

To determine if a relation is a **function**, there are some fundamental criteria to consider:

• Each input \(x\) from the domain must be associated with exactly one output \(y\) from the range.
• If any \(x\) value has more than one corresponding \(y\) value, the relation fails to be a function.

In the context of the exercise, the table you analyzed had the same \(x\) value, 30, leading to multiple different \(y\) values (2, 4, 6, 8, and 10). This clearly demonstrates that the relation is not a function.
Functions are crucial in algebra because they establish a unique, singular relationship between variables. Identifying whether a relation is a function allows us to better understand the behavior of equations and inequalities in algebra.