Select the correct answer.

Jenny is tracking the monthly sales totals for her boutique. The given piecewise function represents the boutique's monthly sales, in dollars, where [tex]\( x \)[/tex] represents the number of months since Jenny began tracking the data.

[tex]\[
f(x) = \left\{
\begin{array}{ll}
4,000(1.1)^x, & 0 \leq x \ \textless \ 3 \\
100x + 5,024, & 3 \leq x \ \textless \ 6 \\
-x^2 + 5x + 5,630, & 6 \ \textless \ x \leq 8
\end{array}
\right.
\][/tex]

What were the boutique's monthly sales when Jenny first began tracking the data?

A. 35,324
B. [tex]\(\$ 4,400\)[/tex]
C. [tex]\(\$ 4,000\)[/tex]
D. [tex]\(\$ 5,616\)[/tex]



Answer :

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.