To determine the correct piecewise function that models the cost [tex]\( f(x) \)[/tex] of [tex]\( x \)[/tex] pounds of trout, we need to consider both the per-pound cost and the shipping costs, based on the quantity Karen orders.
We'll break down the cost according to the amount she orders:
1. For orders less than 10 pounds:
- The cost per pound of trout is [tex]$32.
- The shipping fee is $[/tex]4.
- Therefore, for [tex]\( 0 < x < 10 \)[/tex]:
[tex]\[
f(x) = 32x + 4
\][/tex]
2. For orders of 10 pounds or more:
- The cost per pound of trout drops to [tex]$25.
- The shipping fee increases to $[/tex]9.
- Therefore, for [tex]\( x \geq 10 \)[/tex]:
[tex]\[
f(x) = 25x + 9
\][/tex]
Combining these two conditions, the piecewise function that models the cost [tex]\( f(x) \)[/tex] of trout can be written as:
[tex]\[
f(x) = \left\{
\begin{array}{ll}
32x + 4, & 0 < x < 10 \\
25x + 9, & x \geq 10
\end{array}
\right.
\][/tex]
Now, comparing this result with the given options:
A. [tex]\( f(x) = \left\{ \begin{array}{ll}
25x + 9, & 0 < x < 10 \\
32x + 4, & x \geq 10
\end{array} \right. \)[/tex]
B. [tex]\( f(x) = \left\{ \begin{array}{ll}
32x + 4, & 0 < x < 10 \\
25x + 9, & x \geq 10
\end{array} \right. \)[/tex]
D. [tex]\( f(x) = \left\{ \begin{array}{ll}
25x + 9, & 0 < x \leq 10 \\
32x + 4, & x > 10
\end{array} \right. \)[/tex]
Option B matches our derived piecewise function. Hence, the correct piecewise function is:
[tex]\[ f(x) = \left\{ \begin{array}{ll}
32x + 4, & 0 < x < 10 \\
25x + 9, & x \geq 10
\end{array} \right. \][/tex]
Thus, the correct choice is B.
