Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\begin{array}{|c|c|c|c|c|} \hline x & 11 & 12 & 13 & 14 \\ \hline y & -6 & -6 & -7 & -6 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Domain: \( \{ 11, 12, 13, 14 \} \); Range: \( \{ -7, -6 \} \). Yes, it is a function.

Step 1: Identify the Domain

The domain of a relation is the set of all possible input values (or x-values) for which the relation is defined. From the table, the x-values given are 11, 12, 13, and 14. Thus, the domain \( D \) is \( \{ 11, 12, 13, 14 \} \).

Step 2: Identify the Range

The range of a relation is the set of all possible output values (or y-values) produced by the domain. From the table, the y-values are -6, -6, -7, and -6. The distinct y-values in this relation are -6 and -7. Therefore, the range \( R \) is \( \{ -7, -6 \} \).

Step 3: Determine if the Relation is a Function

A relation is a function if every element in the domain corresponds to exactly one element in the range. From the table, each x-value (11, 12, 13, 14) maps to only one y-value (-6, -6, -7, -6 respectively). Since no x-value corresponds to more than one y-value, the relation is a function.

Key Concepts

Understanding Domain and Range

When working with functions and relations, it's crucial to grasp the definitions of domain and range. The **domain** of any relation refers to the complete set of possible

• input values, or x-values.
• In simpler terms, it's "what we can put into our function."

From the original exercise, we identified the domain by listing out the x-values from the table: 11, 12, 13, and 14.
That leads us to determine the domain as:

• \[D = \{11, 12, 13, 14\}\]

On the other hand, the **range** is all the possible results we obtain.

• This refers to output values, or y-values, after applying the function.
• It is essentially "what comes out of the function."

From the table presented in the exercise, we find y-values as -6, -6, -7, and -6.
Only distinct values count towards the range:

• \[R = \{-6, -7\}\]

Deciphering Relations

Relations occur when we pair inputs with corresponding outputs. Think of a relation as a collection of ordered pairs.

• Each pair consists of one x-value and its corresponding y-value.
• For example, from the table we have pairs like (11, -6) and (13, -7).

Knowing whether a relation is a **function** is the next step.
Functions are special types of relations where:

• Each x-value (input) pairs with only one unique y-value (output).
• This creates a consistent output for any given input, following a predictable pattern.

In the exercise, you can see each x-value, like 12, is uniquely paired with -6.
Hence, this means the relation we're dealing with is a function. No duplicates or erratic pairings exist for x-values here.
Thus, reinforcing its status as a function.

Step by Step Solution Explained

The step by step solution provides clarity and a logical structure for breaking down the problem.
Let's overview this method to understand and demystify the process of finding a solution.**Step 1: Identify the Domain**

• Look for all x-values presented in the table.
• These values form the domain \[D = \{11, 12, 13, 14\}\]

**Step 2: Identify the Range**

• Take note of all y-values, focusing on only unique ones.
• This results in the range \[R = \{-6, -7\}\]

**Step 3: Determine if the Relation is a Function**

• Check each domain element (x-value) to ensure it only maps to one y-value.
• This confirms if the set is a function.

Completing these steps verifies the relation's function nature, guiding us to understand how each x groups with exactly one y.
This simple method leads to immediate understanding and interpretation of any relation or function you might encounter.