To solve this question, let's carefully analyze Andrew's cell phone plan and how the charges are structured. We'll then determine which of the given piecewise functions correctly represents this plan.
1. Understand the Cost Structure:
- For the first 300 minutes, Andrew pays a flat rate of \[tex]$19, regardless of how many minutes he uses, as long as it is within or equal to 300 minutes.
- For any minutes over the initial 300 minutes, Andrew incurs an additional charge of \$[/tex]0.39 per minute.
2. Translating the Cost Structure into a Piecewise Function:
Let's define [tex]\( x \)[/tex] as the total number of minutes Andrew uses in a month.
- For [tex]\( x \leq 300 \)[/tex]: Andrew uses 300 or fewer minutes. The cost is a flat rate of [tex]$19. Therefore, the function for this condition will be:
\[
f(x) = 19
\]
- For \( x > 300 \): Andrew uses more than 300 minutes. He still pays the base amount of $[/tex]19 for the first 300 minutes, plus an additional [tex]$0.39 for each minute over 300. If \( x \) is the total number of minutes used, then the number of minutes over 300 is \( x - 300 \). Therefore, the function for this condition will be:
\[
f(x) = 19 + 0.39(x - 300)
\]
3. Combining the Conditions:
Putting both conditions together into a single piecewise function gives us:
\[
f(x)= \begin{cases}
19, & \text{if } x \leq 300 \\
19 + 0.39(x - 300), & \text{if } x > 300
\end{cases}
\]
4. Verify the Options:
Now let's compare this derived function with the given options:
- Option A: \[
f(x)=\begin{cases}
19, & \text{if } x \leq 300 \\
19 + 0.39x, & \text{if } x > 300
\end{cases}
\]
- This option mistakenly charges an additional \$[/tex]0.39 for all minutes over 300, not just those exceeding 300 minutes.
- Option B: [tex]\[
f(x)=\begin{cases}
19, & \text{if } x > 300 \\
19 + 0.39x, & \text{if } x \leq 300
\end{cases}
\][/tex]
- This option incorrectly swaps the conditions for [tex]\( x \leq 300 \)[/tex] and [tex]\( x > 300 \)[/tex].
- Option C: [tex]\[
f(x)=\begin{cases}
19, & \text{if } x \leq 300 \\
0.39x, & \text{if } x > 300
\end{cases}
\][/tex]
- This option omits the initial \$19 charge for minutes over 300.
- Option D: [tex]\[
f(x)=\begin{cases}
19, & \text{if } x \leq 300 \\
19 + 0.39(x - 300), & \text{if } x > 300
\end{cases}
\][/tex]
- This option correctly represents the cost structure as derived above.
Therefore, the correct piecewise function representing Andrew's cell phone plan is:
[tex]\[
\boxed{D}
\][/tex]
