Let's determine and analyze the piecewise function for Andrew's cell phone plan:
1. When [tex]\( x \leq 300 \)[/tex]:
- When Andrew uses up to 300 minutes in a month ([tex]\( x \leq 300 \)[/tex]), he is charged a flat rate of [tex]$19. This is represented by the function \( f(x) = 19 \).
2. When \( x > 300 \):
- When Andrew uses more than 300 minutes in a month (\( x > 300 \)), he is charged $[/tex]19 for the initial 300 minutes plus [tex]$0.39 for each additional minute.
- The number of additional minutes beyond the first 300 is \( x - 300 \).
- The cost for these additional minutes is \( 0.39 \times (x - 300) \).
- Therefore, the total charge in this case is \( 19 + 0.39 \times (x - 300) \).
Putting both parts together, the piecewise function that represents Andrew's cell phone charges is:
\[ f(x) = \left\{ \begin{array}{ll}
19 & \text{if } x \leq 300 \\
19 + 0.39(x - 300) & \text{if } x > 300
\end{array} \right. \]
Now, let's compare this function with the provided options:
A. \( f(x) = \left\{ \begin{array}{ll} 19, x > 300 \\ 19 + 39x, x \leq 300 \end{array} \right\} \)
- This option is incorrect because for \( x \leq 300 \), the charge should be flat $[/tex]19, not [tex]\( 19 + 39x \)[/tex].
B. [tex]\( f(x) = \left\{ \begin{array}{ll} 19, x \leq 300 \\ 19 + 39x, x > 300 \end{array} \right\} \)[/tex]
- This option is incorrect because for [tex]\( x > 300 \)[/tex], the structure [tex]\( 19 + 39x \)[/tex] does not properly account for the per-minute charge [tex]\( 0.39(x - 300) \)[/tex].
C. [tex]\( f(x) = \left\{ \begin{array}{ll} 19, x \leq 300 \\ 39x, x > 300 \end{array} \right\} \)[/tex]
- This option is incorrect because [tex]\( 39x \)[/tex] does not properly incorporate the flat rate of [tex]$19 and the additional charge for minutes over 300.
D. \( f(x) = \left\{ \begin{array}{ll} 19, x \leq 300 \\ 19 + 39(x - 300), x > 300 \end{array} \right\} \)
- This option correctly reflects the flat rate of $[/tex]19 for up to 300 minutes and the added charge of $0.39 per minute for minutes beyond 300.
Hence, the correct piecewise function representing the charges based on Andrew's cell phone plan is:
[tex]\[ \text{D. } f(x) = \left\{ \begin{array}{ll}
19, & x \leq 300 \\
19 + 0.39(x - 300), & x > 300
\end{array} \right. \][/tex]
