Problem 64 - College Algebra

Question

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

Step-by-Step Solution

Answer

Yes, \( y \) is a function of \( x \).

Step 1: Understand the Definition of a Function

A relation represents \( y \) as a function of \( x \) if each input \( x \) is related to exactly one output \( y \). This means for every \( x \) value, there must be only one corresponding \( y \) value.

Step 2: Analyze the Relationship Between \( x \) and \( y \)

Examine the given table to see if each \( x \) value has a unique \( y \) value. The table rows are (5, 3), (10, 8), and (15, 8).

Step 3: Check for Repeated \( x \) Values with Different \( y \) Values

Look for any \( x \) values that appear more than once with different \( y \) values. In this table, each \( x \) value (5, 10, 15) appears only once and they have corresponding \( y \) values (3, 8, 8) respectively.

Step 4: Confirm if \( y \) is a Function of \( x \)

Since every \( x \) value maps to exactly one \( y \) value (even if different \( x \) values can map to the same \( y \) value), \( y \) is indeed a function of \( x \).

Key Concepts

Relation

In mathematics, a relation is a set of ordered pairs, typically connecting an element from one set, like \( x \), to an element from another set, such as \( y \). Think of it as a mapping between two different sets. When we write these pairing as ordered pairs, the first element is often considered the input, and the second element as the output. For instance, in our table, each number in the top row (5, 10, 15) can be linked to a number in the bottom row (3, 8, 8). This connection of numbers forms a relation.

• A relation might associate one input with one or more outputs, or multiple inputs with a single output.
• In a function, however, each input must associate with exactly one output.

Understanding relations is crucial before diving into the specifics of functions. It's like setting the stage before the main act.

Input-Output Analysis

Input-output analysis in mathematical functions involves understanding how input values (these could be thought of as questions) link to output values (the answers). The table provided in the exercise is a perfect example of this kind of analysis:

• The x-values (5, 10, 15) stand for inputs.
• The corresponding y-values (3, 8, 8) are the outputs.
• When you look at the table, you see how each input has a specific output.

This analysis helps us grasp whether each input relates to only one output, which is key in determining if a relation is a function. By checking how inputs and outputs pair, we can uphold the principles of a function.

Unique Mapping

Unique mapping is an essential concept for defining a function. In the mathematical world, a mapping is just a fancy name for a relationship where inputs connect to outputs. When we talk about a unique mapping, we mean that each input has its own special output, just like a one-to-one correspondence. To determine if the provided numbers in the table have a unique mapping:

• Look at each input \( x \) (5, 10, 15) and see if it links to a single output \( y \) (3, 8, 8).
• In this scenario, no single \( x \) value is paired with multiple \( y \) values, indicating a unique mapping.

This unique mapping verifies that the relation in the table shows \( y \) as a function of \( x \).

Definition of a Function

According to the definition, a function is a special type of relation where each input value (\( x \)) is linked to exactly one output value (\( y \)). This means no single input can have two different outputs.The table from the exercise illustrates this concept well:

• Each \( x \) value—5, 10, and 15—connects to only one \( y \) value.
• Even when different \( x \) values (like 10 and 15) share the same \( y \) value (8), the definition of a function is not violated.

This scenario confirms that \( y \) is indeed a function of \( x \) because every \( x \) is paired uniquely with a single \( y \). Understanding functions helps in various mathematical applications, from simple calculations to complex calculus problems.