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A cost function in mathematics and economics represents the cost of producing goods or services, dependent on various factors like production volume or input costs. In the given exercise, the cost function is specialized for calculating the total cost of shipping a package from one city to another based on the package's weight.

To elaborate, the initial cost for the first pound (or part of it) is a fixed amount, in this case, \( \(10.75 \). For every additional pound or portion thereof, there is an incremental increase in cost, here \( \)3.95 \). This real-world scenario is effectively modeled by a piecewise function because the cost changes in defined steps rather than varying continuously. The function given, \( C = 10.75 + 3.95 \llbracket x \rrbracket \), is a neat representation of the step-wise nature of shipping costs.

When visualizing or calculating costs, it's crucial to consider up to, but not including, the base weight since any additional weight, no matter how small, moves the cost to the next step. This aspect is particularly important to comprehend when predicting expenses or optimizing for cost efficiency.

Step functions are mathematical functions that increase or decrease in steps, rather than varying smoothly. They are also known as staircase functions due to the graphical representation that resembles the steps of a staircase.

In the context of the provided exercise, the cost function \( C = 10.75 + 3.95 \ llbracket x \rrbracket, x > 0 \) acts as a step function because it increases in constant increments each time the weight of the package crosses another pound threshold. The notation \( \llbracket x \rrbracket \) is used to denote the greatest integer less than or equal to \( x \), which ensures that the function increases only at integer values of \( x \), corresponding to full or partial pounds.

Graphing such functions involves creating a series of flat, horizontal segments that suddenly jump to the next value. This visual representation helps users easily identify the cost at any given weight and notice the increment points. While the mathematical concept is straightforward, understanding step functions is vital in industries like shipping and manufacturing, where pricing models often follow a similar pattern.

Graphing functions is an essential skill in mathematics that helps visualize relationships between variables. The graph translates a function's equation into a visual format, making it easier to understand and interpret. In the problem at hand, graphing the step function provides a clear way to comprehend how the cost increases at each additional pound.

To graph the given piecewise function, one should start at the base cost for zero weight, then at each integer increase along the x-axis - indicative of each additional or partial pound - the cost on the y-axis should rise accordingly. It's important to graph open circles at the end of each interval to denote that the function does not include that value but jumps up to the next. Immediately after, filled circles indicate the starting point of the next interval.

This form of graphing is exceptionally practical for visual learners as it helps tie the abstract concept of the function to a concrete visual depiction. It provides immediate insight into how any additional weight will affect the total cost, which is invaluable for budgeting purposes or for making decisions related to shipping costs.