Let's analyze the piecewise function [tex]\( f(x) \)[/tex] and calculate the corresponding [tex]\( y \)[/tex]-values for each given [tex]\( x \)[/tex]-value in the tables provided:
The piecewise function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[
f(x) = \left\{
\begin{array}{ll}
-\frac{5}{4} x + 90, & \text{for } 0 \leq x < 40 \\
-\frac{3}{8} x + 75, & \text{for } 40 \leq x \leq 200
\end{array}
\right.
\][/tex]
### 1. First Table
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
0 & 90 \\
\hline
16 & 85 \\
\hline
40 & 75 \\
\hline
\end{tabular}
\][/tex]
Calculations:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -\frac{5}{4}(0) + 90 = 90
\][/tex]
- For [tex]\( x = 16 \)[/tex]:
[tex]\[
f(16) = -\frac{5}{4}(16) + 90 = -20 + 90 = 70 \quad \text{(This doesn't match the given table value of 85)}
\][/tex]
Since [tex]\( x = 16 \)[/tex] yields 70, not 85, this table is incorrect.
### 2. Second Table
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
0 & 90 \\
\hline
40 & 40 \\
\hline
200 & 0 \\
\hline
\end{tabular}
\][/tex]
Calculations:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -\frac{5}{4}(0) + 90 = 90
\][/tex]
- For [tex]\( x = 40 \)[/tex]:
[tex]\[
f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \quad \text{(This doesn't match the given table value of 40)}
\][/tex]
Since [tex]\( x = 40 \)[/tex] yields 60, not 40, this table is incorrect.
### 3. Third Table
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
40 & 75 \\
\hline
120 & 30 \\
\hline
200 & 0 \\
\hline
\end{tabular}
\][/tex]
Calculations:
- For [tex]\( x = 40 \)[/tex]:
[tex]\[
f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \quad \text{(This doesn't match the given table value of 75)}
\][/tex]
Since [tex]\( x = 40 \)[/tex] yields 60, not 75, this table is also incorrect.
### 4. Fourth Table
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
40 & 60 \\
\hline
160 & 15 \\
\hline
200 & 0 \\
\hline
\end{tabular}
\][/tex]
Calculations:
- For [tex]\( x = 40 \)[/tex]:
[tex]\[
f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60
\][/tex]
- For [tex]\( x = 160 \)[/tex]:
[tex]\[
f(160) = -\frac{3}{8}(160) + 75 = -60 + 75 = 15
\][/tex]
- For [tex]\( x = 200 \)[/tex]:
[tex]\[
f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0
\][/tex]
Based on these calculations, the fourth table correctly reflects the piecewise function values:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
40 & 60 \\
\hline
160 & 15 \\
\hline
200 & 0 \\
\hline
\end{tabular}
\][/tex]
Thus, the correct table to use is the fourth one.
