A car sharing service offers a $50 per month membership fee that includes 10 hours of driving each month and charges $8 for each additional hour. Complete parts (A) through (C).
(A) Write the piecewise definition of the cost [tex]F(x)[/tex] for a month in which a member uses a car for [tex]x[/tex] hours.
[tex]F(x)=\left\{\begin{array}{ll} \square & \text { if } 0 \leq x \leq 10 \\ \square & \text { if } x\ \textgreater \ 10 \end{array}\right.[/tex]
Answer (1)
For 0 { l e q } x { l e q } 10 , the cost is the fixed monthly fee: F ( x ) = 50 .
For 10"> x > 10 , the cost is the monthly fee plus $8 for each additional hour: F ( x ) = 50 + 8 ( x − 10 ) .
Simplify the expression for 10"> x > 10 : F ( x ) = 8 x − 30 .
The piecewise function is: 10 \end{cases}"> F ( x ) = { 50 if 0 ≤ x ≤ 10 8 x − 30 if x > 10 .
Explanation
Understanding the Problem We are given a car sharing service with a monthly fee of $50 that includes 10 hours of driving. If a member drives for more than 10 hours, they are charged $8 for each additional hour. We want to write a piecewise function that describes the cost, F ( x ) , for a month in which a member uses the car for x hours.
Cost for 10 Hours or Less If the member uses the car for 10 hours or less (i.e., 0 { l e q } x { l e q } 10 ), the cost is simply the monthly fee, which is $50.
Cost for More Than 10 Hours If the member uses the car for more than 10 hours (i.e., 10"> x > 10 ), the cost is the monthly fee plus the additional charge for the extra hours. The number of additional hours is x − 10 , and the cost for these hours is 8 ( x − 10 ) . Therefore, the total cost is 50 + 8 ( x − 10 ) . Let's simplify this expression:
50 + 8 ( x − 10 ) = 50 + 8 x − 80 = 8 x − 30
Final Piecewise Function So, the piecewise function is:
10 \end{cases}"> F ( x ) = { 50 if 0 ≤ x ≤ 10 8 x − 30 if x > 10
Examples
Piecewise functions are useful in many real-world scenarios, such as calculating income taxes based on different income brackets or determining shipping costs based on weight. For example, a shipping company might charge a flat rate for packages up to a certain weight and then an additional fee per pound for heavier packages. Understanding piecewise functions helps in modeling and analyzing such situations effectively.
