To determine the boutique's monthly sales when Jenny first began tracking the data, we need to evaluate the piecewise function [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex].
The piecewise function is given by:
[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
For [tex]\( 0 \leq x < 3 \)[/tex], the function is [tex]\( f(x) = 4,000(1.1)^x \)[/tex].
Since Jenny began tracking the data at [tex]\( x = 0 \)[/tex], we substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = 4,000(1.1)^0
\][/tex]
We know that any number raised to the power of 0 is 1, so:
[tex]\[
(1.1)^0 = 1
\][/tex]
Thus, substituting into the equation:
[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.
The correct answer is:
A. \$[/tex]4,000
