To determine what the boutique's monthly sales were when Jenny first began tracking the data, we need to evaluate the piecewise function [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex].
Given the piecewise function:
[tex]\[
f(x)=\left\{\begin{array}{ll}
4,000(1.1)^x, & 0 \leq x<3 \\
100 x+5,024, & 3 \leq x<6 \\
-x^2+5 x+5,630, & 6 < x \leq 8
\end{array}\right.
\][/tex]
When Jenny began tracking the data, [tex]\( x = 0 \)[/tex]. Since [tex]\( 0 \leq x < 3 \)[/tex], we use the first piece of the function:
[tex]\[
f(x) = 4,000(1.1)^x
\][/tex]
Substituting [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 4,000(1.1)^0
\][/tex]
Any number to the power of 0 is 1, so:
[tex]\[
(1.1)^0 = 1
\][/tex]
Thus:
[tex]\[
f(0) = 4,000 \times 1 = 4,000
\][/tex]
Therefore, the boutique's monthly sales when Jenny first began tracking the data were:
[tex]\[
\$4,000
\][/tex]
The correct answer is:
C. [tex]\(\$4,000\)[/tex]
