Problem 27 - 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. $$ \\{(3,4),(3,-4),(4,3),(4,-3)\\} $$

Step-by-Step Solution

Answer

The relation does not define a function because \(x=3\) has two different \(y\) values: 4 and -4.

Step 1: Understanding the Definition of a Function

A relation defines a function if each input value (\(x\)) relates to exactly one output value (\(y\)). In simpler terms, no two ordered pairs should have the same \(x\) value with different \(y\) values.

Step 2: Analyzing the Given Relation

We are given the set of ordered pairs \(\{(3,4),(3,-4),(4,3),(4,-3)\}\). We need to check if any \(x\) value repeats with different \(y\) values.

Step 3: Checking for Repeated x-values

Look at the \(x\) values in the pairs: in the first two pairs (\(3,4\) and \(3,-4\)), the \(x\) value is 3, repeated with different \(y\) values of 4 and -4. This means for \(x=3\), there are multiple \(y\) values.

Step 4: Conclusion

Since the \(x\) value 3 is associated with two different \(y\) values, the relation does not define \(y\) as a function of \(x\).

Key Concepts

Understanding Relations

In mathematics, a relation is a fundamental concept that involves the pairing of elements from one set with elements of another set. This often involves pairing elements of set A with elements of set B. An easy way to visualize a relation is by using ordered pairs. Relations can tell us a lot about how different elements are connected or associated with each other. Here are a few points:

• Relations can consist of any number of ordered pairs.
• Relations are not limited to numbers—they can involve objects, people, or other entities.
• The order in which elements appear in the pairs is significant.
• Understanding relations is crucial as they form the basis for more complex concepts, such as functions and mappings.

Knowing what relations are helps in identifying how variables in mathematics (or objects in reality) may relate to one another. This foundational knowledge is necessary when exploring further topics, especially when dealing with functions.

What Are Ordered Pairs?

Ordered pairs are a simple yet powerful way to define the relationship between two elements. An ordered pair consists of two elements, typically written in the form \((x, y)\), where 'x' is considered the first element or the input, and 'y' is the second element or the output.

• The order of the elements matters significantly. For instance, \((3, 4)\) is not the same as \((4, 3)\).
• Ordered pairs are commonly used in coordinate systems to specify points on a graph.
• They are essential in defining and analyzing relations and functions, as each pair represents a specific output for a given input.

In the context of functions, ordered pairs help determine whether a relation specifies a function or not. They are the building blocks that demonstrate how inputs relate to outputs and serve as the first step in understanding mathematical interactions between different sets.

Decoding Mathematical Definitions

Mathematical definitions provide precise meaning to concepts and ensure clarity in communication. If you're determining whether a relation is a function, understanding the definition of a function is crucial.

• A function is a specific type of relation that links every element in the domain to exactly one element in the range.
• A relation where an input can correspond to multiple outputs is not a function. This is expressed mathematically as "each \(x\) must have one unique \(y\)."
• Re-examining definitions can clarify misunderstandings and provide a clear pathway to analyzing complex problems.

In the problem set at hand, mathematical definitions help us conclude whether the given set of ordered pairs constitutes a function. By checking the definition, we see that a repeated \(x\) with different \(y\) values (as with \((3,4)\) and \((3,-4)\)), indicates it's not a function. Proper understanding of definitions demystifies mathematical concepts and aids problem-solving.