Problem 65 - College Algebra

Question

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

Step-by-Step Solution

Answer

The relation does not represent \(y\) as a function of \(x\) because \(x=10\) corresponds to both \(y=8\) and \(y=14\).

Step 1: Identify Unique Values of x

Evaluate the given table to identify the unique values of the independent variable \(x\). In this table, \(x\) values are 5, 10, and 10.

Step 2: Check for Duplicates in x

Check if any value of \(x\) is repeated in the table. Here, the value \(x = 10\) is repeated, which needs closer examination according to the definition of a function.

Step 3: Compare y Values for Duplicate x

For a relation to represent \(y\) as a function of \(x\), each \(x\) must correspond to exactly one \(y\). Examine the repeated \(x\) value: \(x = 10\) corresponds to two different \(y\) values, 8 and 14.

Step 4: Draw a Conclusion

Since the same \(x\) value (\(x=10\)) corresponds to different \(y\) values, 8 and 14, the relation does not meet the criteria for \(y\) being a function of \(x\).

Key Concepts

Relation and Functions

In algebra, the concepts of relations and functions are foundational. A relation is a simple connection between a set of inputs and a set of outputs. In simpler terms, it matches some inputs to outputs. For example, the table in our exercise shows a relation between values of \(x\) and \(y\). Every relationship or pairing from the input \(x\) to the output \(y\) in the table is part of this broader concept of relation.

Now, a function is a specific type of relation. For a relation to qualify as a function, each input \(x\) must match to exactly one output \(y\). Imagine a vending machine: when you press a button for chips, you don't expect it to give you soda instead. Similarly, in a function, an input should lead to one consistent output. This distinction is critical in understanding both mathematical concepts and real-world scenarios.

Function Criteria

Determining whether a relation is a function involves checking it against specific criteria. The most crucial rule is that every input value \(x\) should be connected to only one output value \(y\). This is what we call the function criteria.

In the example from our exercise, we observe a table with inputs \(x = 5, 10, 10\) and outputs \(y = 3, 8, 14\). Here, \(x\) has been repeated with the value 10. For \(x = 10\), we see that it maps to both \(y = 8\) and \(y = 14\). This violates the function criteria, as the input \(x = 10\) corresponds to more than one output value. Therefore, the relation in the table is not a function.

Functions must be consistent in their mapping to ensure that each input has a single output, keeping the relation predictable and reliable.

Independent and Dependent Variables

The terms 'independent variables' and 'dependent variables' often come up in discussions about functions. These terms help clarify the roles different variables play in relationships and functions.

The independent variable, represented by \(x\) in most functions, is the input or the variable we can change freely. It stands alone and isn't affected by other variables in the function.

On the other hand, the dependent variable, denoted by \(y\), is the output. Its value depends on the value of \(x\). In our given table, \(y\) changes as the \(x\) values change, showing this dependency.

• For \(x = 5\), \(y = 3\).
• For \(x = 10\), \(y = 8\), or \(14\).

This setup implies that if we know the independent variable, we can find the dependent one, provided the relation is a proper function. Understanding these variables helps in constructing and analyzing many mathematical and real-life relationships.