Problem 79 - A Graphical Approach to Precalculus with Limits

Question

The following table lists the U.S. total revenue generated in 2016 in billions of dollars for various companies. $$\begin{array}{l|c}\hline \text { Company } & \text { Revenue } \\\\\hline \text { Apple } & 216 \\\\\text { Alphabet } & 90 \\\\\text { Google } & 89 \\\\\text { Microsoft } & 85\end{array}$$ (a) Using ordered pairs, write a function \(A\) that gives the revenue for each company in billions of dollars. Interpret the first ordered pair. (b) Repeat part (a) using a diagram. (c) Identify the domain and range of \(A\).

Step-by-Step Solution

Answer

(a) \(A = \{(\text{Apple}, 216), (\text{Alphabet}, 90), (\text{Google}, 89), (\text{Microsoft}, 85) \}\); (b) Diagram uses arrows; (c) Domain: \{Apple, Alphabet, Google, Microsoft\}, Range: \{216, 90, 89, 85\}.

Step 1: Create Ordered Pairs for Function A

Each company name will be paired with its revenue to form ordered pairs. The function \( A \) will be written as a set of these pairs: \[A = \{ (\text{Apple}, 216), (\text{Alphabet}, 90), (\text{Google}, 89), (\text{Microsoft}, 85) \}.\] The first ordered pair, \((\text{Apple}, 216)\), represents that the revenue generated by Apple in 2016 is 216 billion dollars.

Step 2: Represent Function A with a Diagram

To visually represent the function \( A \), draw circles to represent each company on the left side and their respective revenues on the right side. Then, draw arrows from each company to its corresponding revenue. For example, draw an arrow from 'Apple' to '216'.

Step 3: Identify the Domain of A

The domain of the function \( A \) consists of all the inputs in the ordered pairs, which are the company names. Thus, the domain is: \( \{ \text{Apple}, \text{Alphabet}, \text{Google}, \text{Microsoft} \} \).

Step 4: Identify the Range of A

The range of the function \( A \) consists of all the outputs in the ordered pairs, which are the revenue figures. Thus, the range is: \( \{ 216, 90, 89, 85 \} \).

Key Concepts

Ordered Pairs

In mathematics, ordered pairs are a fundamental concept. They are used to show relationships between two elements. An ordered pair is written in the form \( (x, y) \), where \( x \) is the first element, known as the 'input,' and \( y \) is the second element, known as the 'output.' For instance, in our exercise about company revenues, each ordered pair represents a company and its corresponding revenue. Here's why ordered pairs are interesting:

• The order matters: \( (x, y) \) is different from \( (y, x) \).
• They can represent various relationships such as coordinates on a graph, matches in a tournament, or in our case, companies and their revenues.

In the function \( A = \{ (\text{Apple}, 216), (\text{Alphabet}, 90), (\text{Google}, 89), (\text{Microsoft}, 85) \}\), the first ordered pair \((\text{Apple}, 216)\) tells us that in 2016, Apple generated 216 billion dollars. Understanding ordered pairs is key because they help in representing and analyzing data in a structured manner.

Domain and Range

When discussing functions, two important sets we often refer to are the domain and range. They tell us about the inputs and outputs of the function.

• Domain: The set of all possible "inputs" for the function. In other words, it includes the first elements of the ordered pairs.
• Range: The set of all possible "outputs" for the function. This means the second elements from each ordered pair.

In our example, the domain of the function \( A \) includes all company names: \( \{ \text{Apple}, \text{Alphabet}, \text{Google}, \text{Microsoft} \} \). These are the entities for which revenue data is available. The range is given by the revenue numbers: \( \{ 216, 90, 89, 85 \} \), which are the values generated by each company.
The domain and range help in understanding the full picture of what the function covers, and knowing these sets is crucial for defining the function's behavior.

Function Diagram

A function diagram is a visual representation that helps illustrate the relationship between inputs and outputs in a function. It simplifies the understanding of how each element in the domain corresponds to an element in the range. Here’s how it works:

• Draw two sets of circles or ovals: one set on the left side representing the domain (inputs) and another on the right side for the range (outputs).
• Use arrows to connect elements that are related. An arrow originates from an input and points to its corresponding output.
  For example, an arrow from 'Apple' on the left points to '216' on the right, indicating that Apple’s revenue is 216 billion dollars.

The function diagram for our scenario provides an easy-to-understand map of the associations, allowing one to grasp the function's pattern visually. It's particularly helpful in identifying distinct or repetitive relationships and can be an effective way to check if each input is linked correctly to just one output, a crucial property of functions. A function diagram is not always necessary but can be indispensable when dealing with complex data sets.