Problem 31 - 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. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 7 \\ 2 & 15 \\ 3 & 23 \\ 4 & 16 \\ 5 & 8 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The relation is a function, as each \(x\) is paired with only one \(y\) value.

Step 1: Understand the Problem

We need to determine if the given set of ordered pairs expresses a function. A relation can be called a function if each input value (\(x\)) maps to exactly one output value (\(y\)). If a single \(x\) value corresponds to multiple \(y\) values, the relation is not a function.

Step 2: Analyze Each Pair

We are given the pairs \((1, 7), (2, 15), (3, 23), (4, 16), (5, 8)\). We need to check if each \(x\) value is associated with exactly one \(y\) value.

Step 3: Check for Repeated x-values

Go through each pair and check if an \(x\) value repeats with a different \(y\) value. If an \(x\) value is used more than once with different \(y\) values, then it is not a function. Here, each \(x\) value is unique: 1, 2, 3, 4, and 5 are all different and each maps to only one \(y\).

Step 4: Conclusion

Since each \(x\) value is paired with exactly one \(y\) value and no \(x\) is repeated with different \(y\) values, this means the relation is a function.

Key Concepts

Ordered Pairs

In mathematics, ordered pairs represent a fundamental concept where two elements are paired together in a specific sequence. The general form of an ordered pair is \(x, y\), where \(x\) is often termed as the first component or the input, and \(y\) as the second component or the output. The order in which these components appear is crucial.

• The first element (usually \(x\)) typically represents an input or independent variable.
• The second element (usually \(y\)) represents an output or dependent variable.

It's important to remember that in an ordered pair, the sequence is always fixed, making \( (x, y) \) different from \( (y, x) \). In a mathematical context, especially when discussing functions or relations, the order ensures a clear definition of how inputs map to outputs.

Relation in Mathematics

A relation in mathematics is essentially a collection of ordered pairs. It describes how certain elements from one set (usually the domain) relate to elements in another set (usually the codomain). When considering a set of ordered pairs, such as \((1, 7), (2, 15), (3, 23), (4, 16), (5, 8)\), the underlying relationship connects each \(x\) value to its corresponding \(y\) value.

• A relation can contain multiple pairs, and each pair indicates that there is some form of connection between the \(x\) and \(y\) values.
• Not all relations are functions, but all functions are specific types of relations.

In the context of determining whether a relation represents a function, we often check if each \(x\) is unique when paired with a \(y\). If any \(x\) value links to more than one \(y\), the relation isn't considered a function. This uniqueness condition is vital for defining functions confidently.

Function Definition

In simple terms, a function is a special type of relation where each distinct input is associated with exactly one output. This one-to-one mapping between each \(x\) (input) to a single unique \(y\) (output) differentiates functions from more general relations. Mathematically, we can formalize this as a condition:

• For the relation to be a function, no two ordered pairs can have the same first element \(x\) with different second elements \(y\). In our given pairs \((1, 7), (2, 15), (3, 23), (4, 16), (5, 8)\), each \(x\) is distinct and maps to a unique \(y\).
• This makes it clear that the relation indeed forms a function, as each x-value appears once and maps to only one y-value.

Understanding functions is crucial as they are frequently used to represent real-world situations mathematically, where a consistent, predictable relationship is needed between variables. Functions offer this reliability by ensuring that each input will always produce the same output.