Problem 32 - A Graphical Approach to College Algebra

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. $$\begin{array}{c|c|c|c|c} \boldsymbol{x} & 1 & 1 & 1 & 1 \\ \hline \boldsymbol{y} & 12 & 13 & 14 & 15 \end{array}$$

Step-by-Step Solution

Answer

Domain: \( \{ 1 \} \), Range: \( \{ 12, 13, 14, 15 \} \), Not a function.

Step 1: Determine the Domain

The domain of a relation is the set of all possible input values (x-values) from the given points. In this table, the input value \( x \) is consistently 1. Therefore, the domain \( D \) is \( \{ 1 \} \).

Step 2: Determine the Range

The range of a relation is the set of all possible output values (y-values) from the given points. Here, the y-values are 12, 13, 14, and 15. Thus, the range \( R \) is \( \{ 12, 13, 14, 15 \} \).

Step 3: Check if the Relation is a Function

A relation is a function if every input is associated with exactly one output. Here, the input \( x = 1 \) has multiple outputs (12, 13, 14, and 15). Therefore, this relation is not a function because a single input does not map to a single output.

Key Concepts

Relations and Functions

When we talk about relations in algebra, we are simply referring to a set of ordered pairs. A relation can be represented as a table, a graph, or even an equation. Relations show how two sets of information are related: one as the input (usually denoted as \( x \)) and the other as the output (usually denoted as \( y \)).

A function is a special type of relation. In a function, each input is paired with exactly one output. It's like a special machine that always gives you the same answer when you put in the same question.

To determine if a relation is a function, ensure that no input \( x \) has more than one corresponding output \( y \). If you see an input related to multiple outputs, it’s not a function—just like in our exercise example, where input \( x = 1 \) is paired with outputs 12, 13, 14, and 15.

Input and Output Values

Understanding input and output values is crucial to working with relations and functions. The input value is what you provide to a relation or function as an \( x \)-value. The output value is the result you get after applying the relation or function, represented by the \( y \)-value.

In algebra, inputs are usually the independent variable, and outputs are the dependent variable. The independent variable is what you control, and the dependent variable depends on your input.

• In our exercise, the input or \( x \)-value is always 1, which makes it easy to find the domain.
• The output or \( y \)-values are 12, 13, 14, and 15, making up the range.

In summary, whenever tackling a problem, identify your inputs and outputs first. This will help you understand what the relation or function is doing and see if multiple outputs map to a single input, which helps determine if it's a function.

Table Representation in Algebra

Tables are a clear and organized way to represent relations in algebra. One column typically represents all inputs, while another column represents the outputs. This method allows you to see the relationship between \( x \)-values and \( y \)-values quickly.

For our exercise, the table shows:

• The input \( x \) repeated as 1.
• The outputs \( y \) as 12, 13, 14, and 15.

By displaying data in a table, it's easier to identify patterns and relationships at a glance, such as repeated inputs or varied outputs.

It's a great way to visually check if a relation is a function: remember, a function will not have the same input \( x \) corresponding to different \( y \)-values. Using tables not only clarifies information but also aids in accurately drawing conclusions about the nature of the relation or function.