Let's carefully analyze the cell phone plan and determine which piecewise function correctly represents it.
Tracy’s cell phone plan details:
- Tracy pays [tex]$29 for up to 250 minutes.
- If Tracy uses more than 250 minutes, she pays an additional $[/tex]0.35 per minute for each minute over 250.
We need to identify the piecewise function that correctly captures these charges.
### Step-by-Step Analysis
1. Basic Charge:
- For usage [tex]\(x \leq 250\)[/tex] minutes:
- Tracy pays the flat rate of [tex]$29.
- This implies the function should return 29 when \(x \leq 250\).
2. Additional Charge:
- For usage \(x > 250\) minutes:
- Tracy’s total charge includes the flat rate of $[/tex]29, plus she pays [tex]$0.35 for every minute over 250 minutes.
- The additional cost would be calculated as follows:
- If \(x\) is the total number of minutes used,
- The extra minutes used over 250 would be \(x - 250\).
- The additional charge for these extra minutes is \(0.35 (x - 250)\).
- Therefore, the total charge is \(29 + 0.35(x - 250)\).
### Formulating the Piecewise Function
Based on the analysis, we can write the piecewise function as:
\[ f(x) = \begin{cases}
29 & \text{if } x \leq 250 \\
29 + 0.35(x - 250) & \text{if } x > 250
\end{cases} \]
### Matching to the Given Options
We compare our derived piecewise function with the given options:
- \(f(x)= \left\{ \begin{array}{c} 29, x \leq 250 \\ 29+0.35 x, x > 250 \end{array} \right\}\)
- This option incorrectly calculates the charge for \(x > 250\) as \(29 + 0.35x\), which does not account for the 250 free minutes.
- \(f(x)= \left\{ \begin{array}{c} 29, x \leq 250 \\ 29+0.35(x-250), x > 250 \end{array} \right\}\)
- This option correctly represents the charges for both scenarios:
- For \(x \leq 250\), it charges $[/tex]29.
- For [tex]\(x > 250\)[/tex], it charges [tex]$29 + 0.35(x - 250)$[/tex], considering the additional minutes over 250.
- [tex]\(f(x)= \left\{ \begin{array}{c} 29, x > 250 \\ 29+0.35 x, x \leq 250 \end{array} \right\}\)[/tex]
- This option misplaces the conditions and does not reflect the cell phone plan accurately.
- [tex]\(f(x)= \left\{ \begin{array}{c} 29, x \leq 250 \\ 0.35 x, x > 250 \end{array} \right\}\)[/tex]
- This option again incorrectly calculates the charge for [tex]\(x > 250\)[/tex] as [tex]\(0.35x\)[/tex], failing to include the base fee of $29.
The correct piecewise function that represents Tracy's cell phone plan is:
[tex]\[ f(x) = \left\{ \begin{array}{c} 29, x \leq 250 \\ 29 + 0.35(x - 250), x > 250 \end{array} \right\} \][/tex]
Therefore, the correct answer is the second option.
