Problem 34 - 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. $$\begin{array}{c|c|c|c|c|c} \boldsymbol{x} & 1 & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} & \frac{1}{16} \\\ \hline \boldsymbol{y} & 0 & -1 & -2 & -3 & -4 \end{array}$$

Step-by-Step Solution

Answer

Domain: \(\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}\}\); Range: \(\{0, -1, -2, -3, -4\}\); It is a function.

Step 1: Identify the Domain

The domain of a relation consists of all the possible input values, which are the values listed for \(x\). Here, the \(x\) values are \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}\). Therefore, the domain \(D\) is \(\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}\}\).

Step 2: Identify the Range

The range of a relation consists of all possible output values, which are the values listed for \(y\). The \(y\) values here are \(0, -1, -2, -3, -4\). So, the range \(R\) is \(\{0, -1, -2, -3, -4\}\).

Step 3: Determine if the Relation is a Function

A relation is a function if each input value \(x\) corresponds to exactly one output value \(y\). In the given table, every \(x\) value maps to a unique \(y\) value. Thus, this relation is a function.

Key Concepts

Understanding Domain and Range

When we talk about the domain and range in mathematics, we're referring to essential concepts that describe the input and output of a function or relation. The **domain** is all the possible input values (usually represented as "x") that a relation can accept. It's like the set of ingredients you can use in a recipe. For example, if you have the inputs \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}\), then those make up your domain.

The **range**, on the other hand, involves all possible output values (usually represented as "y") that can be generated by the relation. Think of the range as the list of possible dishes you can prepare with your set of ingredients. In this case, our outputs or range values are \(0, -1, -2, -3, -4\).

Understanding these concepts is crucial when dealing with relations and functions since it determines what values we can input and what we can expect as outcomes.

Relations in Mathematics

In mathematics, a **relation** is essentially a connection between different sets of values. It defines how elements from one set (the input) relate to elements of another set (the output). Relations can be visualized using mapping diagrams, graphs, or tables that link specific inputs to certain outputs.

A relation doesn’t necessarily follow any specific rule for connections between inputs and outputs. This makes relations a broader concept compared to functions. When looking at our table:

• Inputs: \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}\)
• Outputs: \(0, -1, -2, -3, -4\)

We can see how each input value relates to an output value. Identifying such connections helps in forming a comprehensive understanding of whether a relation can be expressed more specifically as a function.

Identifying Functions

Identifying whether a relation is a **function** is a key skill in mathematics. A function is a specific type of relation where each input value maps to exactly one output value. This is similar to a vending machine – you press the button for a snack, and it gives you one snack. You wouldn’t expect it to give you two different snacks for the same button press.

To establish that a relation is a function, follow these rules:

• Check that each input in your list has one and only one output.
• No input should be connected to more than one output.

In our table:

• Each \(x\) value \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}\) is associated with a unique \(y\) value \(0, -1, -2, -3, -4\) respectively.

Since there is a unique y for every x, the relation described here is indeed a function. Understanding this helps pinpoint whether a rule or pattern exists in the relation, making it more predictable and systematic.