Let's analyze Tracy's cell phone plan step by step to determine the correct piecewise function that represents her monthly charges.
Given:
1. A flat rate of [tex]$29 for up to 250 free minutes.
2. Any additional minutes over the 250 free minutes are charged at $[/tex]0.35 per minute.
We'll break this down into cases based on the number of minutes [tex]\(x\)[/tex] Tracy uses:
Case 1: [tex]\( x \leq 250 \)[/tex]:
If Tracy uses 250 minutes or fewer in a month, no extra charges are incurred. Therefore, the total cost is simply the flat rate:
[tex]\[ f(x) = 29 \][/tex]
for [tex]\(x \leq 250\)[/tex].
Case 2: [tex]\( x > 250 \)[/tex]:
If Tracy uses more than 250 minutes in a month, she will incur additional charges for the extra minutes beyond the 250 free minutes. Specifically:
- The cost for the first 250 minutes is still the flat rate of [tex]$29.
- For the additional minutes beyond 250, the cost is $[/tex]0.35 per minute.
Let's denote the additional minutes beyond 250 as [tex]\(x - 250\)[/tex]. The additional cost for these minutes would be:
[tex]\[ 0.35 \times (x - 250) \][/tex]
Therefore, the total cost in this case becomes:
[tex]\[ f(x) = 29 + 0.35 \times (x - 250) \][/tex]
for [tex]\( x > 250 \)[/tex].
Putting both cases together, the piecewise function representing Tracy's monthly charges is:
[tex]\[ f(x) = \begin{cases}
29 & \text{if } x \leq 250 \\
29 + 0.35 \times (x - 250) & \text{if } x > 250
\end{cases} \][/tex]
Comparing this with the given options, the correct piecewise function is:
C. [tex]\( f(x) = \left\{\begin{array}{c}
29, x \leq 250 \\
29 + 0.35(x - 250), x > 250
\end{array}\right\} \)[/tex]
So the answer is C.
