To solve this problem, we need to clearly define the charges based on Andrew's cell phone plan using a piecewise function.
Let's break down Andrew's cell phone plan:
1. If Andrew uses up to 300 minutes in a month, he only pays a flat rate of [tex]$19.
2. For every minute over 300, Andrew pays an additional $[/tex]0.39 per minute.
Now let's look at the options given to see which one matches this scenario.
### Step-by-step Analysis of Each Option
#### Option A
[tex]\[ f(x) = \begin{cases}
19 & \text{if } x > 300 \\
19 + 39x & \text{if } x \leq 300
\end{cases} \][/tex]
- This option states that if Andrew uses more than 300 minutes, the cost will be [tex]$19. However, this is incorrect because for more than 300 minutes, additional charges apply.
- For \( x \leq 300 \), the function claims the cost is \( 19 + 39x \), which is incorrect because the flat rate of $[/tex]19 applies, not a multiplication.
#### Option B
[tex]\[ f(x) = \begin{cases}
19 & \text{if } x \leq 300 \\
19 + 0.39(x - 300) & \text{if } x > 300
\end{cases} \][/tex]
- For [tex]\( x \leq 300 \)[/tex], the cost is correctly given as [tex]$19.
- For \( x > 300 \), the cost is calculated as the base $[/tex]19 plus [tex]$0.39 for each minute over 300, which is indeed \( 19 + 0.39(x - 300) \).
This option looks correct, let's check the rest.
#### Option C
\[ f(x) = \begin{cases}
19 & \text{if } x \leq 300 \\
19 + 39x & \text{if } x > 300
\end{cases} \]
- For \( x \leq 300 \), the cost is correctly $[/tex]19.
- For [tex]\( x > 300 \)[/tex], the function says the cost is [tex]\( 19 + 39x \)[/tex], which incorrectly multiplies the extra minutes by 39 instead of multiplying by 0.39 per extra minute.
#### Option D
[tex]\[ f(x) = \begin{cases}
19 & \text{if } x \leq 300 \\
39x & \text{if } x > 300
\end{cases} \][/tex]
- For [tex]\( x \leq 300 \)[/tex], the cost is [tex]$19 which is correct.
- For \( x > 300 \), the function says the cost is \( 39x \), which implies multiplying all the minutes by 39, ignoring the initial 300 minutes of $[/tex]19.
### Conclusion
Option B correctly reflects the structure of the cell phone plan. For [tex]\( x \leq 300 \)[/tex], the cost is [tex]$19, and for \( x > 300 \), the cost is the base $[/tex]19 plus $0.39 for each minute over 300. Therefore, the correct piecewise function that represents the charges based on Andrew's cell phone plan is:
[tex]\[ f(x) = \begin{cases}
19 & \text{if } x \leq 300 \\
19 + 0.39(x - 300) & \text{if } x > 300
\end{cases} \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
