Let's analyze each of the given options to understand which one correctly represents Tracy's cell phone plan, where she gets up to 250 free minutes for a flat rate of [tex]$29, and is charged $[/tex]0.35 for every minute over 250.
We need to determine the correct piecewise function:
1. Option A:
[tex]\[
f(x)=\left\{\begin{array}{c}
29, \quad x \leq 250 \\
29 + 0.35x, \quad x > 250
\end{array}\right\}
\][/tex]
In this option:
- For [tex]\(x \leq 250\)[/tex], it charges [tex]$29 for up to 250 minutes, which is correct.
- For \(x > 250\), it calculates the cost as \(29 + 0.35x\). This incorrectly charges 0.35 for all minutes, including the first 250 minutes which should be free.
Therefore, this option is incorrect.
2. Option B:
\[
f(x)=\left\{\begin{array}{c}
29, \quad x > 250 \\
29 + 0.35x, \quad x \leq 250
\end{array}\right\}
\]
In this option:
- For \(x > 250\), it charges $[/tex]29, regardless of extra minutes, which is incorrect.
- For [tex]\(x \leq 250\)[/tex], it charges [tex]\(29 + 0.35x\)[/tex], which is also incorrect because it implies paying more than [tex]$29 for fewer than 250 minutes.
Therefore, this option is incorrect.
3. Option C:
\[
f(x)=\left\{\begin{array}{c}
29, \quad x \leq 250 \\
29 + 0.35(x - 250), \quad x > 250
\end{array}\right\}
\]
In this option:
- For \(x \leq 250\), it correctly charges $[/tex]29 for up to 250 minutes.
- For [tex]\(x > 250\)[/tex], it charges [tex]$29 plus $[/tex]0.35 for each minute over 250. The term [tex]\(0.35(x - 250)\)[/tex] properly accounts for only the minutes over 250.
Therefore, this option is correct.
4. Option D:
[tex]\[
f(x)=\left\{\begin{array}{c}
29, \quad x \leq 250 \\
0.35x, \quad x > 250
\end{array}\right\}
\][/tex]
In this option:
- For [tex]\(x \leq 250\)[/tex], it charges [tex]$29, which is correct.
- For \(x > 250\), it incorrectly charges \(0.35x\) without including the initial $[/tex]29 flat rate. The initial $29 charge must be included.
Therefore, this option is incorrect.
After reviewing all options, the correct piecewise function representing the charges based on Tracy's cell phone plan is:
[tex]\[
c. \quad f(x)=\left\{\begin{array}{c}
29, \quad x \leq 250 \\
29 + 0.35(x - 250), \quad x > 250
\end{array}\right\}
\][/tex]
