Problem 82 - A Graphical Approach to College Algebra

Question

The following table lists Square's daily transactions \(y\) in millions of dollars, \(x\) months after March 2011 . (Square can be used to accept credit cards on your iPhone.) $$\begin{array}{c|c} x & y \\ \hline 0 & 1.0 \\ 2 & 2.0 \\ 7 & 5.5 \\ 12 & 11.0 \end{array}$$ (a) Using ordered pairs, write a function \(T\) that gives the daily transactions in millions of dollars during each month. Interpret the first ordered pair. (b) Repeat part (a) using a diagram. (c) Identify the domain and range of \(T\).

Step-by-Step Solution

Answer

Function: \(T(x)\), Domain: \{0, 2, 7, 12\}, Range: \{1.0, 2.0, 5.5, 11.0\}.

Step 1: Represent Data as Ordered Pairs

Let's write the given data as ordered pairs. Each pair \(x, y\) consists of \(x\), the number of months after March 2011, and \(y\), the daily transactions in millions of dollars.\[ (0, 1.0), (2, 2.0), (7, 5.5), (12, 11.0) \]The first ordered pair, (0, 1.0), tells us that at 0 months after March 2011, the daily transactions were 1.0 million dollars.

Step 2: Create a Function from Ordered Pairs

Using the ordered pairs, we can define a function \(T(x)\) where \(T\) gives the daily transactions for any month \(x\):\[ T(x) = \begin{cases} 1.0, & x = 0 \2.0, & x = 2 \5.5, & x = 7 \11.0, & x = 12 \end{cases} \]This function \(T\) represents the given transactions for each corresponding month after March 2011.

Step 3: Illustrate Using a Mapping Diagram

In a mapping diagram, each element from the set of inputs \(x\) is connected to the corresponding output \(T(x)\).- Input Values: \(0, 2, 7, 12\)- Output Values: \(1.0, 2.0, 5.5, 11.0\)The connections are:- 0 maps to 1.0- 2 maps to 2.0- 7 maps to 5.5- 12 maps to 11.0

Step 4: Determine Domain and Range of Function

The domain of the function \(T\) is the set of all possible inputs, i.e., the months after March 2011 for which transactions are given:\[ \text{Domain} = \{0, 2, 7, 12\} \]The range of the function \(T\) is the set of all transaction values corresponding to the months in the domain:\[ \text{Range} = \{1.0, 2.0, 5.5, 11.0\} \]

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics used to define relationships between two quantities. An ordered pair is composed of two elements: the first element is often an input, and the second is its corresponding output. It is written as \( (x, y) \). In the context of the given exercise, the ordered pairs represent the time in months after March 2011 and the corresponding daily transactions in millions of dollars.
\[ (0, 1.0), (2, 2.0), (7, 5.5), (12, 11.0) \]
The first number in each pair (like 0 in the pair (0, 1.0)) represents the month, while the second number (such as 1.0) symbolizes daily transactions.
Understanding ordered pairs helps to easily identify the relationship between variables, often represented as \(x \) and \(y \). Each pair provides a snapshot of what the relationship looks like at a specific point.
When considering functions, these ordered pairs are inputs and outputs of the function itself. This is a simple yet effective way to represent data that can show trends or evaluate functions like \(T(x)\), which highlights the transactions for each month.

Domain and Range

The terms domain and range are essential when discussing functions. In simple words, the domain is the set of all possible input values, and the range is the set of all possible output values.
How does this apply to our exercise? The domain here refers to the months after March 2011, for which Square's transactions are recorded.
\[ \text{Domain} = \{0, 2, 7, 12\} \]

This means there are four specific months included in this dataset.
On the flip side, the range is concerned with the scope of output values—transactions for those months.
\[ \text{Range} = \{1.0, 2.0, 5.5, 11.0\} \]
Having the domain and range defined allows you to know precisely what the function covers and its possible outcomes, giving a clearer picture of how the function \(T(x)\) operates.

Mapping Diagram

A mapping diagram offers a visual way to express the relationships between inputs and outputs in a function. This diagram consists of two columns. The first column has the input values, while the second comprises the output values.
Each line or arrow drawn between the columns shows a connection from an input ( {x} ) to the output ( {T(x)} ).

• 0 maps to 1.0
• 2 maps to 2.0
• 7 maps to 5.5
• 12 maps to 11.0

This kind of illustration helps in understanding how each input is linked with its corresponding output. It's a straightforward method to get an overview of a function and is particularly useful in recognizing patterns or ensuring every input has an output. This clarity provided by a mapping diagram supports students in grasping concepts like functions and helps in visual learning.