Andrew has a cell phone plan that provides 300 free minutes each month for a flat rate of $19. For any minutes over 300, Andrew is charged $0.39 per minute. Which of the following piecewise functions represents charges based on Andrew's cell phone plan?
A. [tex]f(x)=\left{\begin{array}{c}19, x\ \textgreater \ 300 \\ 19+.39 x, x \leq 300\end{array}\right}[/tex]
B. [tex]f(x)=\left{\begin{array}{c}19, x \leq 300 \\ 19+.39(x-300), x\ \textgreater \ 300\end{array}\right}[/tex]
C. [tex]f(x)=\left{\begin{array}{c}19, x \leq 300 \\ .39 x, x\ \textgreater \ 300\end{array}\right}[/tex]
D. [tex]f(x)=\left{\begin{array}{c}19, x \leq 300 \\ 19+.39 x, x\ \textgreater \ 300\end{array}\right}[/tex]
Answer (1)
Define x as the number of minutes Andrew uses.
For x ≤ 300 , the charge is $19.
For 300"> x > 300 , the charge is 19 + 0.39 ( x − 300 ) .
The piecewise function is 300 \end{cases}"> f ( x ) = { 19 , x ≤ 300 19 + 0.39 ( x − 300 ) , x > 300 , so the answer is B .
Explanation
Define the variable Let x be the number of minutes Andrew uses in a month. We need to determine the piecewise function that represents the charges based on Andrew's cell phone plan.
Charges for 300 minutes or less If Andrew uses 300 minutes or less, i.e., x ≤ 300 , the charge is a flat rate of $19.
Charges for more than 300 minutes If Andrew uses more than 300 minutes, i.e., 300"> x > 300 , the charge is $19 plus $0.39 for each minute over 300. This can be expressed as $19 + 0.39(x - 300).
Write the piecewise function Therefore, the piecewise function f ( x ) can be written as: 300 \end{cases}"> f ( x ) = { 19 , x ≤ 300 19 + 0.39 ( x − 300 ) , x > 300
Select the correct option Comparing the derived piecewise function with the given options, we see that option B matches our result.
Final Answer The correct piecewise function that represents Andrew's cell phone plan is: 300 \end{cases}"> f ( x ) = { 19 , x ≤ 300 19 + 0.39 ( x − 300 ) , x > 300 Therefore, the answer is B.
Examples
Piecewise functions are useful in modeling situations where different rules or conditions apply over different intervals. For example, consider a parking garage that charges a flat fee for the first two hours and then an hourly rate for each additional hour. This can be modeled using a piecewise function, where one piece represents the flat fee for the initial hours, and another piece represents the hourly rate for the subsequent hours. Understanding piecewise functions helps in analyzing and predicting costs in such scenarios.
