Problem 65 - College Algebra with Corequisite Support

Question

For the following exercises, determine if the relation represented in table form represents \(y\) as a function of \(x\). $$ \begin{array}{|c|c|c|c|} \hline x & 5 & 10 & 10 \\ \hline y & 3 & 8 & 14 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

No, the relation does not represent \( y \) as a function of \( x \) because \( x = 10 \) maps to two different \( y \) values (8 and 14).

Step 1: Understand the Definition of a Function

A relation between two variables, usually denoted as \( y \) and \( x \), is considered a function if every input \( x \) is associated with exactly one output \( y \). In simpler terms, each \( x \) value should map to only one \( y \) value.

Step 2: Examine the Table Values

Look at each \( x \) value in the table. The values are \( 5, 10, \) and \( 10 \). Observe the corresponding \( y \) values: 3, 8, and 14. Notice that the \( x \) value 10 is repeated, but it has two different corresponding \( y \) values (8 and 14).

Step 3: Apply Function Criteria

According to the function criteria from Step 1, a function cannot have an \( x \) value that maps to more than one \( y \) value. Here, since \( x = 10 \) has two different \( y \) values (\( y = 8 \) and \( y = 14 \)), this does not satisfy the condition for \( y \) to be a function of \( x \).

Step 4: Determine the Conclusion

Since the \( x \) value of 10 does not have a unique \( y \) value, we conclude that this relation does not represent \( y \) as a function of \( x \).

Key Concepts

Relations

In mathematics, a relation between two sets is a collection of ordered pairs, where the first element of each pair comes from the first set (called the domain), and the second element comes from the second set (called the range). Each connection or pairing between elements of these sets is what we refer to as a relation. For example, in a relation such as a table or a graph, each such pairing corresponds to a link that defines some kind of interaction or dependency between the variables.

Relations can be vast and complex, ranging from simple lists of pairs to intricate maps describing real-world scenarios. It's important to note that while all functions are relations, not all relations are functions. This is due to the specific criteria that functions must meet, which will be detailed later. Understanding the concept of relations serves as the foundation to grasp more specific topics like functions.

Input-Output Pairs

Every relation is formed by input-output pairs where each pair typically consists of a value from the domain (input) and a corresponding value from the range (output). In the exercise example, the x-values (5, 10, 10) serve as inputs, while the y-values (3, 8, 14) are outputs.
These pairs can be visualized in tables, graphs, or may simply be listed. The method of presentation does not change the fundamental nature of the relation.

Recognizing input-output pairs is crucial for determining if a relation is a function. In fact, this identification process lays down the framework for understanding more complex mathematical concepts where input-output processes are observed, such as calculus and statistical analysis. In studying such pairs, repetition of an input value with different outputs signals that the relation might not meet the criteria of a function.

Function Criteria

For a relation to be classified as a function, it must meet a specific criterion: each input must be associated with exactly one output. In simpler terms, a function from a set of inputs (domain) to a set of outputs (range) requires that no two outputs correspond to the same input. Think of it this way: an input is like a key, and each key must unlock exactly one value.

In the given exercise, the input value 10 is paired with both 8 and 14. This immediately disqualifies it from being a function since it doesn’t adhere to the exclusive one-to-one correspondence needed between inputs and outputs.

When examining relations to determine if they are functions, focus first on the inputs: check if any are repeated with different outputs. If so, the relation does not meet the function criteria. Understanding this concept helps when dealing with different representations of functions, whether they’re illustrated as equations, graphs, or tables.