An express-mail company charges 25 for a package weighing...

An express-mail company charges $\$ 25$ for a package weighing up to 2 pounds. For each additional pound or fraction of a pound, there is an additional charge of $\$ 3 .$ Let $D(x)$ represent the cost to send a package weighing $x$ pounds. Graph $y=D(x)$ for $x$ in the interval $(0,6]$

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Key Concepts

Piecewise Functions

Piecewise functions are defined by using different expressions over different parts of their domain. In many real-world applications, such as cost calculations, the behavior of the function changes at certain thresholds, requiring a different rule for each interval.

Step Functions

Step functions are a type of piecewise function that take on constant values over specific intervals and then jump to different constant values at the endpoints of those intervals. They model systems where there are sudden changes, such as a fixed cost increase after a threshold is exceeded.

Ceiling Function (Rounding Up)

The ceiling function rounds any number up to the nearest whole number. This concept is important when the cost increments occur for every additional unit or fraction thereof, meaning even a partial unit is charged as if it were a full unit.

Graphing Discontinuous Functions

Graphing functions that are piecewise or step in nature involves drawing horizontal segments for the portions where the function remains constant and clearly marking jump discontinuities at the boundaries between different intervals. Understanding how to represent open and closed endpoints is crucial in accurately depicting the function's behavior.

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