Problem 7 - Elementary and Intermediate Algebra

Question

U.S. Recycling. The following table gives the approximate number of aluminum cans (in billions) collected each year for the years \(2000-2006\). a. Display the data in the table as a relation, that is, as a set of ordered pairs. b. Find the domain and range of the relation. c. Use an arrow diagram to show how members of the range correspond to members of the domain. $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \begin{array}{l} \text { Billions of } \\ \text { aluminum cans } \end{array} & 63 & 56 & 54 & 50 & 52 & 51 & 51 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Domain: \(\{2000, 2001, 2002, 2003, 2004, 2005, 2006\}\); Range: \(\{63, 56, 54, 50, 52, 51\}\).

Step 1: Create Ordered Pairs

The task involves representing the data as a set of ordered pairs where each pair consists of the year and the number of aluminum cans collected in billions. Write down each pair: \((2000, 63), (2001, 56), (2002, 54), (2003, 50), (2004, 52), (2005, 51), (2006, 51)\).

Step 2: Identify Domain and Range

The domain of a relation is the set of all possible first elements of the ordered pairs, while the range is the set of all possible second elements. Here, the domain is \(\{2000, 2001, 2002, 2003, 2004, 2005, 2006\}\) and the range is \(\{63, 56, 54, 50, 52, 51\}\). Note that the year 2005 and 2006 share the same range value of 51.

Step 3: Construct Arrow Diagram

To create an arrow diagram, list the elements of the domain on the left and the elements of the range on the right. Draw arrows from each domain element to its corresponding range value. This results in arrows from 2000 to 63, 2001 to 56, 2002 to 54, 2003 to 50, 2004 to 52, and both 2005 and 2006 to 51, illustrating the relationship.

Key Concepts

Ordered Pairs

An ordered pair is a simple yet fundamental concept in mathematics, especially when dealing with relations and functions. An ordered pair looks like this: \((a, b)\) where "\(a\)" is known as the first element, and "\(b\)" is the second element. The order is crucial and denotes a relationship between these two elements. For instance, in our recycling data example, an ordered pair such as \((2000, 63)\) represents that in the year 2000, 63 billion aluminum cans were recycled. Each ordered pair captures a snapshot of data, connecting a specific year with the corresponding number of cans collected.

• The "first element" typically represents an input, such as a point in time.
• The "second element" shows an output or the result connected to that input.
• Ordering is important: \((2000, 63)\) is different from \((63, 2000)\).

Ordered pairs are crucial for visualizing relations, allowing us to see how each year is connected with its data point. It's a bit like creating a map between two sets of data. This concept is a building block for determining domains and ranges and for creating visual representations like graphs and arrow diagrams.

Domain and Range

When dealing with relations, understanding the concepts of domain and range is essential. The **domain** of a relation is all the possible values that the first element, usually the "input," can take. In our recycling data, the domain consists of the years provided. Hence, the domain is: \[\{2000, 2001, 2002, 2003, 2004, 2005, 2006\}\]This domain tells us that the data covers these specific years.
The **range**, on the other hand, represents all possible values for the second element, commonly referred to as the "output." Here, the range includes all recorded numbers of cans collected, like so: \[\{63, 56, 54, 50, 52, 51\}\]Notice that "51" appears twice because both the years 2005 and 2006 reported the same number of cans collected. However, in describing a set, we list unique values only.

• The domain is all possible initial values or inputs (here, the years).
• The range is all possible output results associated with the domain values (here, the number of cans).

In this recycling data, understanding domains and ranges helps pinpoint what years and data points are being examined and tracked.

Arrow Diagram

An arrow diagram is a useful graphical representation of a relation, showing connections between elements of the domain and their corresponding elements in the range.
To construct an arrow diagram:

• List all domain elements (here, the years) vertically on the left side.
• Do the same for the range elements (here, the number of cans) on the right side.
• Draw arrows from each element in the domain to its specific corresponding element in the range.

In our aluminum can data example, you'd see arrows from 2000 to 63, 2001 to 56, and so forth. Interestingly, both 2005 and 2006 would have arrows pointing to the number 51, indicating the shared value.
This visual setup helps in understanding the interaction within the data, making it easier to see which inputs are connected to which outputs. It's particularly handy in identifying cases where different inputs lead to the same output, reinforcing concepts of domains and ranges visually.