Problem 63 - College Algebra with Corequisite Support

Question

For the following exercises, determine if the relation represented in table form represents \(y\) as a function of \(x\). $$ \begin{array}{|c|c|c|c|} \hline x & 5 & 10 & 15 \\ \hline y & 3 & 8 & 14 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Yes, the relation is a function because each \(x\) has one unique \(y\).

Step 1: Understand the Table Format

The table given represents a relation between two variables, \(x\) and \(y\). The first row contains the values of \(x\), and the second row contains the corresponding values of \(y\). Each pair of \(x\) and \(y\) values indicates a relationship for that particular data point.

Step 2: Review the Definition of a Function

A function is a relation where each input (\(x\) value) has exactly one output (\(y\) value). This means that in the context of the table, for each \(x\), there can only be one associated \(y\).

Step 3: Examine Each \(x\) Value in the Table

Look at the \(x\) values in the table: 5, 10, and 15. They are all distinct and do not repeat. This indicates that each \(x\) value is paired with exactly one \(y\) value.

Step 4: Analyze the Corresponding \(y\) Values

Review the \(y\) values corresponding to each \(x\): 3 for \(x = 5\), 8 for \(x = 10\), and 14 for \(x = 15\). Each \(x\) maps to one unique \(y\), confirming the definition of a function.

Step 5: Conclude Based on the Analysis

Since every \(x\) value in the table has a unique \(y\) value, the relation given in the table satisfies the definition of a function.

Key Concepts

Relation Between Variables

When we talk about relations in mathematics, we often refer to the connection between two variables. In this exercise, we are dealing with a relation between the variables \(x\) and \(y\). The table showcases values of \(x\) along with their corresponding \(y\) values. A relation simply tells us how the values of one variable are related to the values of another.

For example:

• If \(x\) is 5, \(y\) is 3.
• If \(x\) is 10, \(y\) is 8.
• If \(x\) is 15, \(y\) is 14.

This relation indicates how a specific value of \(x\) matches with a value of \(y\).

Identifying this correctly is crucial as it lays the foundation for understanding further concepts like functions.

Definition of a Function

A function is a special type of relation. For a relation to be a function, every input or \(x\) value must correspond to one and only one output or \(y\) value. This is the crux of understanding functions.

• No \(x\) value can have more than one \(y\) value.
• Each \(x\) must be paired with a unique \(y\).

In the given exercise, each \(x\) value of 5, 10, and 15 pairs with exactly one \(y\) value such as 3, 8, and 14 respectively.

Because every \(x\) is associated with one distinct \(y\), this setup fits the strict definition of a function.

Understanding whether a relation is a function is pivotal in grasping how mathematical models predict outcomes based on specific inputs.

Table Representation of Functions

Tables are a practical way to present functions because they clearly show the relationship between \(x\) and \(y\) values. In our exercise, the table is structured such that:

• The top row lists all possible \(x\) values.
• The bottom row provides the corresponding \(y\) values for each \(x\).

This format makes it easy to analyze whether every \(x\) has one unique \(y\).

A quick glance at the table helps us see why the relation qualifies as a function. No \(x\) value appears more than once, and each has a single, matched \(y\) value. This clear representation also aids in comprehending more complex functional behaviors later in your mathematical journey.

When using tables, always ensure to check for uniqueness in \(y\) values per \(x\) values. This ensures the relation meets the criteria of a function efficiently.