To determine the monthly sales when Jenny first began tracking the data, we need to evaluate the given piecewise function at [tex]\( x = 0 \)[/tex].
Given the piecewise function:
[tex]\[
f(x)=\left\{\begin{array}{ll}
4,000(1.1)^3, & 0 \leq x<3 \\
100 x+5,024, & 3 \leq x<6 \\
-x^2+5 x+5,630, & 6
We need to check which piece of the function applies when [tex]\( x = 0 \)[/tex].
For [tex]\( 0 \leq x < 3 \)[/tex]:
[tex]\[
f(x) = 4,000(1.1)^3
\][/tex]
Because [tex]\( x \)[/tex] is 0 (which falls in the range [tex]\( 0 \leq x < 3 \)[/tex]), we use the first part of the function. Calculating this gives:
[tex]\[
f(0) = 4,000 \times (1.1)^3
\][/tex]
Evaluating the expression:
[tex]\[
1.1^3 = 1.331
\][/tex]
[tex]\[
4,000 \times 1.331 = 5324
\][/tex]
Therefore, the sales when Jenny first began tracking the data were \[tex]$5,324. The correct answer is neither of the provided options (A, B, C, or D), since none of them match \$[/tex]5,324 accurately.
However, sticking to the prompt, if there is any discrepancy in the given choices, we note that Jenny's boutique had sales of [tex]\( \$ 5,324 \)[/tex] when she first began tracking. Unfortunately, it appears none of the given answer choices correctly matches this calculated value.
