To determine which piecewise function accurately represents the charges based on Ben's cell phone plan, let's break down the information given:
1. Ben has 200 free minutes each month for a flat rate of [tex]$39.
2. For any minutes over 200, Ben is charged $[/tex]0.35 per minute.
Now, let's analyze each piecewise function one by one:
1. Option 1:
[tex]\[
f(x)=\left\{\begin{array}{l}
39, \quad x \leq 200 \\
0.35(x-200), \quad x > 200
\end{array}\right\}
\][/tex]
- When [tex]\(x \leq 200\)[/tex], it charges a flat fee of [tex]$39, which is correct.
- When \(x > 200\), it charges $[/tex]0.35 per minute for every minute over 200 but does not include the initial [tex]$39 fee. This is incorrect because it should add this additional cost to the base fee.
2. Option 2:
\[
f(x)=\left\{\begin{array}{l}
39, \quad x > 200 \\
39 + 0.35x, \quad x \leq 200
\end{array}\right\}
\]
- This function charges a flat fee of $[/tex]39 only for [tex]\(x > 200\)[/tex], which is incorrect.
- For [tex]\(x \leq 200\)[/tex], it calculates the charge as [tex]$39 plus $[/tex]0.35 per minute, which also does not match the plan's description. Therefore, it is incorrect.
3. Option 3:
[tex]\[
f(x)=\left\{\begin{array}{l}
39, \quad x \leq 200 \\
39 + 0.35(x-200), \quad x > 200
\end{array}\right\}
\][/tex]
- For [tex]\(x \leq 200\)[/tex], it correctly charges a flat fee of [tex]$39.
- For \(x > 200\), it correctly applies the additional charge to the base fee, calculating it as $[/tex]39 (base fee) + [tex]$0.35 per minute for every minute over 200, which aligns perfectly with the plan description.
4. Option 4:
\[
f(x)=\left\{\begin{array}{l}
39, \quad x \leq 200 \\
0.35x, \quad x > 200
\end{array}\right\}
\]
- For \(x \leq 200\), it charges a flat fee of $[/tex]39, which is correct.
- For [tex]\(x > 200\)[/tex], it charges [tex]$0.35 per minute for all minutes, not just for minutes over 200, and omits the initial $[/tex]39 fee. Therefore, it is incorrect.
After evaluating all the options, we see that the correctly represented piecewise function is:
[tex]\[
f(x)=\left\{\begin{array}{l}
39, \quad x \leq 200 \\
39 + 0.35(x-200), \quad x > 200
\end{array}\right\}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
