Problem 65 - College Algebra with Modeling and Visualization

Question

First-Class Mail In March 2008 , the retail flat rate in dollars for first- class mail weighing up to 5 ounces could be computed by the piece wise- constant function \(P\), where \(x\) is the number of ounces. $$ P(x)=\left\\{\begin{array}{ll} 0.80 & \text { if } 0

Step-by-Step Solution

Answer

(a) P(1.5) = 0.97, P(3) = 1.14. (b) Domain: (0, 5]. (c) Discontinuous at x = 1, 2, 3, 4.

Step 1: Determine P(1.5)

First, we need to determine the value of the function at specific points. For \(P(1.5)\), since \(1 < 1.5 \leq 2\), we use the rule \(P(x) = 0.97\). Thus, \(P(1.5) = 0.97\). This means the cost for mailing a letter weighing 1.5 ounces is $0.97.

Step 2: Determine P(3)

For \(P(3)\), since \(2 < 3 \leq 3\), we use the rule \(P(x) = 1.14\). Thus, \(P(3) = 1.14\). This means the cost for mailing a letter weighing exactly 3 ounces is $1.14.

Step 3: Sketch the Graph of P

To sketch the graph, plot flat segments based on the function definition. Start with the segment from \( (0, 0.80) \) to \( (1, 0.80) \), then \( (1, 0.97) \) to \( (2, 0.97) \), and so forth until the last point \( (5, 1.48) \). Each segment is horizontal, as the price is constant within each interval.

Step 4: Determine the Domain of P

The domain of \(P\) is the interval \(0 < x \leq 5\), which means the function is defined for any letter weight from more than 0 ounces up to and including 5 ounces.

Step 5: Identify Discontinuities

The function \(P\) is discontinuous at the points \(x = 1, 2, 3, 4\), where the function value changes from one constant value to another abruptly.

Key Concepts

First-Class Mail Rate

The retail flat rate for first-class mail is an example of a piecewise function because different rates apply to different weight intervals of the mail. In March 2008, the cost of mailing, measured in ounces, could be determined using specific rate brackets:

• 0.80 for mail weighing greater than 0 but up to 1 ounce
• 0.97 for mail weighing more than 1 but up to 2 ounces
• 1.14 for mail weighing more than 2 but up to 3 ounces
• 1.31 for mail weighing more than 3 but up to 4 ounces
• 1.48 for mail weighing more than 4 but up to 5 ounces

Each bracket is constant, meaning that as long as your mail fits within a specific weight interval, you will pay the same amount. This can be quite useful for customers who can predict and adjust their shipping costs based on the weight of their packages.

Function Discontinuity

Discontinuity in a function occurs where there is an abrupt change in the value of the function. In a piecewise function like the first-class mail rate, discontinuities are common at the boundary points where one rate ends, and another begins. For the first-class mail rate, these points of discontinuity occur at the ounce marks of 1, 2, 3, and 4. At each of these points, the function switches from one constant rate to another.
To visualize this, consider the graph of this function; at each discontinuity point, there is a "jump" from one horizontal line (or rate) to another. This indicates that the cost suddenly changes as soon as the weight crosses into the next interval. Such jumps are characteristic of piecewise constant functions like this one.

Graphing Functions

Graphing a piecewise function like the first-class mail rate involves plotting each interval segment on the coordinate plane. Each interval corresponds to a flat line with a horizontal orientation, reflecting the constant rate within that interval.

• Begin at the point (x = 0, cost = 0.80), and draw a horizontal line to (x = 1, cost = 0.80).
• Next, draw a new horizontal line starting (x = 1, cost = 0.97) up to (x = 2, cost = 0.97).
• Continue this process for each weight interval.
• At each interval's end, there's a gap or "jump" to the start of the next line, which represents the discontinuities.

The domain of this function is from more than 0 ounces to 5 ounces, which are the intervals you'll need to cover on your graph. Observing the flat sections and jumps helps you understand how mailing costs shift with slight changes in package weight.