Speeding Tickets In a certain state the maximum speed...

Speeding Tickets In a certain state the maximum speed permitted on freeways is $65 \mathrm{mi} / \mathrm{h}$, and the minimum is $40 \mathrm{mi} / \mathrm{h}$. The fine $F$ for violating these limits is $\$ 15$ for every mile above the maximum or below the minimum. a. Complete the expressions in the following piece wise defined function, where $x$ is the speed at which you are driving. b. Find $F(30), F(50),$ and $F(75)$ c. What do your answers in part (b) represent?

Speeding Tickets In a certain state the maximum speed permitted on freeways is $65 \mathrm{mi} / \mathrm{h}$, and the minimum is $40 \mathrm{mi} / \mathrm{h}$. The fine $F$ for violating these limits is $\$ 15$ for every mile above the maximum or below the minimum.
a. Complete the expressions in the following piece wise defined function, where $x$ is the speed at which you are driving.
b. Find $F(30), F(50),$ and $F(75)$
c. What do your answers in part (b) represent?

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Key Concepts

Piecewise Functions

Piecewise functions are defined by different expressions over various parts of the domain. In contexts like speed limits and fines, the function adjusts the penalty based on whether the driving speed is above a maximum, below a minimum, or in the allowed range. This concept is essential for modeling scenarios where the behavior of the function changes based on specific conditions.

Function Evaluation

Function evaluation refers to the process of substituting a particular input value into a function's formula to calculate the corresponding output. This is crucial for determining specific outcomes like the amount of a speeding ticket when a driver is traveling at a given speed.

Linear Relationships

Linear relationships describe functions where the change in the dependent variable is directly proportional to the change in the independent variable. In the speeding ticket example, the fine increases linearly with each mile per hour that a driver's speed exceeds the maximum or falls below the minimum, reflecting a straightforward proportional penalty.

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