Problem 63 - College Algebra

Question

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

Step-by-Step Solution

Answer

Yes, the relation represents \(y\) as a function of \(x\).

Step 1: Understand the Definition of a Function

A relation is a function if each input value corresponds to exactly one output value. In this scenario, we need to check if each value of \(x\) is associated with only one \(y\) value.

Step 2: Identify Inputs and Outputs

From the table, the input values are \(x = 5, 10, 15\) and the corresponding output values are \(y = 3, 8, 14\).

Step 3: Check One-to-One Correspondence

In the given table, observe each \(x\) value: 5 is paired with 3, 10 is paired with 8, and 15 is paired with 14. Each \(x\) has a unique \(y\) value, indicating a one-to-one correspondence.

Step 4: Determine if the Relation is a Function

Since each \(x\) value in the table is paired with exactly one \(y\) value, the relation represented by the table is a function.

Key Concepts

Relation and Function

In algebra, understanding the difference between a relation and a function is fundamental. A **relation** refers to any set of ordered pairs, where each pair consists of two elements. In these ordered pairs, the first element is usually an input, and the second element is the corresponding output.

On the other hand, a **function** is a special type of relation. For a relation to be considered a function, each input must be paired with exactly one output. This means that no input can correspond to multiple outputs. For example, if you have input values like \( x = 5, 10, 15 \) and each has a unique output, as in \( y = 3, 8, 14 \), then this relation is indeed a function.

Remember the following:

• A function expresses a unique objective relationship between inputs and outputs.
• If any input is linked to more than one output, the relation is not a function.
• In simple terms, if you put something into a function, you should always get the same thing out.

Understanding whether a relation is a function helps in predicting outputs and is crucial for solving algebraic equations.

One-to-One Correspondence

One-to-one correspondence is a key concept when determining whether a relation is a function. It means there's a perfect pairing between inputs and outputs – each input is matched with one and only one output. This concept ensures that for every \( x \) value there is a unique \( y \) value.

Let's delve deeper:

• In a one-to-one correspondence, there can be no repetition of output values for the same input.
• For example, in our previous table: \( 5 \rightarrow 3 \), \( 10 \rightarrow 8 \), \( 15 \rightarrow 14 \), each \( x \) has one unique \( y \).
• This makes the relation both a function and a bijection (a type of one-to-one matching).

When checking one-to-one correspondence, examine if repeated pairs share identical output values. If even one doesn't, the relation may not be a function. Hence, ensuring one-to-one correspondence confirms that a given set of inputs and outputs form a reliable function.

Input-Output Table

An input-output table is a helpful tool in identifying functions in algebra. It consists of a systematically arranged set of data points representing a relation. Each **input** is aligned with an **output** in a clear format.

This table displays:

• Inputs (usually labeled under \( x \)) on one side.
• Corresponding outputs (under \( y \)) on the other side.

In our example table:

• The inputs are \( x = 5, 10, 15 \).
• The corresponding outputs are \( y = 3, 8, 14 \).

Spotting a function becomes much easier when data is organized in an input-output table. You can quickly check for one-to-one correspondence by looking for any coexistence between inputs and multiple outputs.

Always remember: an input-output table isn't just about aligning values. It's about ensuring clarity, making it possible to identify patterns and establish relationships between variables.