Question

The United States Postal Service first class mail rates are $$\$ 0.41$$ for the first ounce and $$\$ 0.17$$ for each additional ounce or fraction thereof up to \(3.5\) ounces. A model for the cost \(C\) (in dollars) of a first class mailing that weighs \(3.5\) ounces or less is given below. (\mathrm{\\{} S o u r c e : ~ United States Postal Service) \(C(x)=\left\\{\begin{array}{ll}0.41, & 0 \leq x \leq 1 \\ 0.58, & 1

Short answer

The cost function is not continuous at x = 1, x = 2, x = 3. The cost to mail a 2.5-ounce letter is $0.75.

Step-by-step solution

Discuss Continuity

Starting with the first part of the problem, to discuss continuity of this function, visualize the graph. You'll see that the function is not continuous at x = 1, x = 2, x = 3. These are the points where the function 'jumps' from one cost to another. This is because these are the weights at which the postal rate changes.

Graphing the Function

Graph the function using a graphing tool. The y-axis represents the mailing cost and the x-axis represents the weight of the item being mailed. Plot four horizontal line segments for the four price tiers, with each segment starting at the low end of that tier's weight range, up to but not including the next range.

Calculate the Mail Cost

Coming to the second part of the problem, find the cost of mailing a letter that weighs 2.5 ounces. According to the function, when the weight lies between 2 and 3 ounces (2 < x <= 3), the cost is $0.75. So, the cost of mailing a 2.5 ounce letter is $0.75.

Key concepts

Graphing Piecewise Functions

Piecewise functions, like the postal service cost model, are defined by different expressions for different intervals of the input variable.

When graphing piecewise functions, it is crucial to consider each interval and its corresponding rule. For the given problem, there are four distinct intervals each with a flat rate that does not change with weight within that interval. To graph this function, we start by plotting four horizontal segments on the cost (y-axis) against weight (x-axis). Each segment corresponds to a price tier and extends just up to the next tier without touching it, reflecting the different cost per weight interval.

To make these functions easier to understand, consider plotting each piece with a different color or pattern. This visual differentiation helps emphasize the piece-nature of the function. Remember that open or closed circles at the end of each line segment indicate whether the endpoint is included or excluded in the interval.

Continuity in Piecewise Functions

Continuity in a piecewise function is determined by whether there is an unbroken path along the graph of the function. For the cost model in question, we want to check if the graph jumps or has gaps at certain points.

The function provided in the exercise is not continuous at certain points specified by the weight increments, each corresponding to a change in rate. These points are where x equals 1, 2, and 3 ounces. These are considered to be discontinuities because at each instance, the cost jumps to a new value rather than changing gradually. In the graph, these discontinuities are represented by open circles followed immediately by a distinct horizontal segment with a higher y-value. An understanding of continuity is crucial when considering how a function behaves and ensuring that our model accurately reflects real-life scenarios, such as how postal rates are structured.

Applying Piecewise Functions

Applying piecewise functions, including the postal rate function, involves substituting a given input into the correct expression based on the defined intervals.

In the context of our exercise, to compute the cost for mailing a letter that weighs 2.5 ounces, identify the correct interval first. In this case, 2 < x ≤ 3, which corresponds to a cost of $0.75. Understanding the application of the function not only helps in solving textbook problems but also extends to real-world scenarios. For example, determining shipping costs, comparing rates from different suppliers, or calculating fees based on weight categories are all instances where piecewise functions prove useful. Emphasizing the ability to apply piecewise functions can aid students in visualizing mathematics not just as an abstract concept, but as a practical tool.