Problem 30 - 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. $$ \\{(-2,4),(-3,8),(-3,12),(-4,16)\\} $$

Step-by-Step Solution

Answer

The relation is not a function because the x-value -3 has two different y-values, 8 and 12.

Step 1: Understand the Definition of a Function

A relation is a function if for every input value (x) there is exactly one output value (y). This means there can't be two ordered pairs with the same x-value but different y-values.

Step 2: Identify x-value Repetitions

Examine the given pairs: \((-2,4),(-3,8),(-3,12),(-4,16)\). Look for any x-values that are repeated with different y-values.

Step 3: Analyze the x-values

Observe that the x-value \(-3\) is repeated: in the pairs \((-3,8)\) and \((-3,12)\). These pairs show that two different y-values correspond to the same x-value, meaning the relation does not define y as a function of x.

Step 4: State the Conclusion

Since the x-value \(-3\) corresponds to two different y-values (8 and 12), the relation is not a function.

Key Concepts

Relation vs Function

In algebra, understanding the difference between a relation and a function is vital. A relation is simply a set of ordered pairs. It tells us how two elements are related but does not impose restrictions on their correspondence. On the other hand, a function is a special kind of relation where each input value (often represented as \(x\)) has a unique output value (commonly denoted as \(y\)).
This means, in a function:

• For every \(x\), there is exactly one \(y\).
• No \(x\) in the set can pair with multiple \(y\) values.

Understanding this difference allows us to classify and work with different sets of ordered pairs based on whether they fulfill the criteria of a function.

Ordered Pairs

Ordered pairs are a fundamental concept that help describe relationships between two elements. An ordered pair is written in the form \((x, y)\). The first element \(x\) is known as the input or independent variable, while the second element \(y\) is referred to as the output or dependent variable.
These pairs provide the foundation for defining relations and identifying functions. In practical terms, ordered pairs help us map out the connection between two variables, allowing for clear visualization and analysis of possible patterns or rules governing their relationship.

x-values and y-values

Within the context of ordered pairs, the terms \(x\)-values and \(y\)-values are integral. The \(x\)-value represents the independent or input variable, while the \(y\)-value signifies the dependent or output variable.

• The \(x\)-values are the first elements of each ordered pair.
• The \(y\)-values are the second elements.

The relationship between these values is essential for determining whether a given set of pairs represents a function. If each \(x\)-value corresponds to one and only one \(y\)-value throughout, then we have a function. Monitoring how these values interact gives us critical insight into the nature of the relation at hand.

Repeated x-values

One of the key tasks in determining if a relation is a function is to look for repeated \(x\)-values within the set of ordered pairs. In a function, no \(x\) value can repeat unless the corresponding \(y\) value is the same every time.
Consider the ordered pairs: \((-2,4), (-3,8), (-3,12), (-4,16)\). Here, we observe that the \(x\) value \(-3\) is repeated. But critically, it pairs with two different \(y\) values: 8 and 12. This repetition with differing \(y\) values immediately tells us the relation is not a function. By checking for repeated \(x\)-values, we efficiently ascertain the functional quality of a relation, making it a crucial step in algebraic analysis.