To ship small packages within the United States, a shipping...

To ship small packages within the United States, a shipping company charges $\$ 3.75$ for the first pound and $\$ 1.10$ for each additional pound or fraction of a pound. Let $C(x)$ represent the cost of shipping a package, and let $x$ represent the weight of the package. Graph $C(x)$ for any package weighing up to (and including) $6 \mathrm{lb}$.

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Key Concepts

Graphing Piecewise Functions

Graphing a piecewise function involves carefully plotting each segment on its corresponding interval and clearly indicating open or closed endpoints to reflect whether an endpoint is included. This approach is essential for accurately interpreting functions with conditions or discontinuities, especially in cost models.

Piecewise Functions

A piecewise function is one that is defined by different expressions over distinct intervals of the input variable. In a shipping cost problem, the cost formula varies between the initial flat rate and the additional incremental rates, making the function piecewise and modeling the conditions on different weight intervals.

Step Functions

A step function is characterized by constant values within intervals with sudden jumps at the boundaries. In contexts where any fraction of a unit is charged as a full unit, the cost function increases in 'steps', reflecting the discrete nature of additional charges for weight fractions.

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