Answer :
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.
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.
