To determine the boutique's monthly sales when Jenny first began tracking the data, we need to evaluate the given piecewise function at [tex]\( x = 0 \)[/tex].
The piecewise function is:
[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
Since [tex]\( x = 0 \)[/tex] falls within the range [tex]\( 0 \leq x < 3 \)[/tex], we use the first part of the piecewise function:
[tex]\[
f(x) = 4,000(1.1)^x
\][/tex]
Substitute [tex]\( x = 0 \)[/tex] into this expression:
[tex]\[
f(0) = 4,000 (1.1)^0
\][/tex]
Since any number raised to the power of 0 is 1:
[tex]\[
(1.1)^0 = 1
\][/tex]
Therefore:
[tex]\[
f(0) = 4,000 \times 1 = 4,000
\][/tex]
So, the boutique's monthly sales when Jenny first began tracking the data were [tex]$4,000.
Thus, the correct answer is:
B. $[/tex]4,000
