Answer :
To find the tables that contain the correct pairs of input and output values for the function [tex]\( f(x) = \frac{3 + 5x}{2} \)[/tex], we will verify each table by calculating the function for each input value and checking it against the given outputs.
Let's start with each table and verify the correctness of each input-output pair step-by-step.
Table 1:
[tex]\[ \begin{tabular}{|l|r|r|r|r|} \hline Input ( $x$ ) & -1.6 & 0.4 & 2.2 & 3.7 \\ \hline Output $f(x)$ & -5.5 & 2.5 & 6.75 & 10.75 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = -1.6 \)[/tex]:
[tex]\[ f(-1.6) = \frac{3 + 5(-1.6)}{2} = \frac{3 - 8}{2} = \frac{-5}{2} = -2.5 \][/tex]
Given output is -5.5, which is incorrect.
Table 2:
[tex]\[ \begin{tabular}{|l|l|l|l|r|} \hline Input $(x)$ & -2.7 & -1.3 & 0.8 & 4.4 \\ \hline Output $f(x)$ & -5.25 & -1.75 & 3.5 & 12.5 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = -2.7 \)[/tex]:
[tex]\[ f(-2.7) = \frac{3 + 5(-2.7)}{2} = \frac{3 - 13.5}{2} = \frac{-10.5}{2} = -5.25 \][/tex]
The output matches.
2. [tex]\( x = -1.3 \)[/tex]:
[tex]\[ f(-1.3) = \frac{3 + 5(-1.3)}{2} = \frac{3 - 6.5}{2} = \frac{-3.5}{2} = -1.75 \][/tex]
The output matches.
3. [tex]\( x = 0.8 \)[/tex]:
[tex]\[ f(0.8) = \frac{3 + 5(0.8)}{2} = \frac{3 + 4}{2} = \frac{7}{2} = 3.5 \][/tex]
The output matches.
4. [tex]\( x = 4.4 \)[/tex]:
[tex]\[ f(4.4) = \frac{3 + 5(4.4)}{2} = \frac{3 + 22}{2} = \frac{25}{2} = 12.5 \][/tex]
The output matches.
Table 3:
[tex]\[ \begin{tabular}{|l|r|r|r|r|} \hline Input ( $x$ ) & 1.1 & 2.4 & 5.3 & 6.9 \\ \hline Output $f(x)$ & 4.25 & 7.5 & 11.75 & 17.25 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = 1.1 \)[/tex]:
[tex]\[ f(1.1) = \frac{3 + 5(1.1)}{2} = \frac{3 + 5.5}{2} = \frac{8.5}{2} = 4.25 \][/tex]
The output matches.
2. [tex]\( x = 2.4 \)[/tex]:
[tex]\[ f(2.4) = \frac{3 + 5(2.4)}{2} = \frac{3 + 12}{2} = \frac{15}{2} = 7.5 \][/tex]
The output matches.
3. [tex]\( x = 5.3 \)[/tex]:
[tex]\[ f(5.3) = \frac{3 + 5(5.3)}{2} = \frac{3 + 26.5}{2} = \frac{29.5}{2} = 14.75 \][/tex]
Given output is 11.75, which is incorrect.
Table 4:
[tex]\[ \begin{tabular}{|l|r|r|r|r|} \hline Input $(x)$ & -2.1 & 0.9 & 1.7 & 3.3 \\ \hline Output $f(x)$ & -3.75 & 3.75 & 5.75 & 9.75 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = -2.1 \)[/tex]:
[tex]\[ f(-2.1) = \frac{3 + 5(-2.1)}{2} = \frac{3 - 10.5}{2} = \frac{-7.5}{2} = -3.75 \][/tex]
The output matches.
2. [tex]\( x = 0.9 \)[/tex]:
[tex]\[ f(0.9) = \frac{3 + 5(0.9)}{2} = \frac{3 + 4.5}{2} = \frac{7.5}{2} = 3.75 \][/tex]
The output matches.
3. [tex]\( x = 1.7 \)[/tex]:
[tex]\[ f(1.7) = \frac{3 + 5(1.7)}{2} = \frac{3 + 8.5}{2} = \frac{11.5}{2} = 5.75 \][/tex]
The output matches.
4. [tex]\( x = 3.3 \)[/tex]:
[tex]\[ f(3.3) = \frac{3 + 5(3.3)}{2} = \frac{3 + 16.5}{2} = \frac{19.5}{2} = 9.75 \][/tex]
The output matches.
Table 5:
[tex]\[ \begin{tabular}{|l|r|r|r|r|} \hline Input $(x)$ & 2.4 & 3.5 & 4.6 & 5.0 \\ \hline Output $f(x)$ & 7.5 & 10.5 & 12.5 & 14 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = 2.4 \)[/tex]:
[tex]\[ f(2.4) = \frac{3 + 5(2.4)}{2} = \frac{3 + 12}{2} = \frac{15}{2} = 7.5 \][/tex]
The output matches.
2. [tex]\( x = 3.5 \)[/tex]:
[tex]\[ f(3.5) = \frac{3 + 5(3.5)}{2} = \frac{3 + 17.5}{2} = \frac{20.5}{2} = 10.5 \][/tex]
The output matches.
3. [tex]\( x = 4.6 \)[/tex]:
[tex]\[ f(4.6) = \frac{3 + 5(4.6)}{2} = \frac{3 + 23}{2} = \frac{26}{2} = 13 \][/tex]
Given output is 12.5, which is incorrect.
4. [tex]\( x = 5.0 \)[/tex]:
[tex]\[ f(5.0) = \frac{3 + 5(5.0)}{2} = \frac{3 + 25}{2} = \frac{28}{2} = 14 \][/tex]
Output is 14, which is correct.
Therefore, the tables that contain correct pairs of input and output values are Table 2 and Table 4.
Let's start with each table and verify the correctness of each input-output pair step-by-step.
Table 1:
[tex]\[ \begin{tabular}{|l|r|r|r|r|} \hline Input ( $x$ ) & -1.6 & 0.4 & 2.2 & 3.7 \\ \hline Output $f(x)$ & -5.5 & 2.5 & 6.75 & 10.75 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = -1.6 \)[/tex]:
[tex]\[ f(-1.6) = \frac{3 + 5(-1.6)}{2} = \frac{3 - 8}{2} = \frac{-5}{2} = -2.5 \][/tex]
Given output is -5.5, which is incorrect.
Table 2:
[tex]\[ \begin{tabular}{|l|l|l|l|r|} \hline Input $(x)$ & -2.7 & -1.3 & 0.8 & 4.4 \\ \hline Output $f(x)$ & -5.25 & -1.75 & 3.5 & 12.5 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = -2.7 \)[/tex]:
[tex]\[ f(-2.7) = \frac{3 + 5(-2.7)}{2} = \frac{3 - 13.5}{2} = \frac{-10.5}{2} = -5.25 \][/tex]
The output matches.
2. [tex]\( x = -1.3 \)[/tex]:
[tex]\[ f(-1.3) = \frac{3 + 5(-1.3)}{2} = \frac{3 - 6.5}{2} = \frac{-3.5}{2} = -1.75 \][/tex]
The output matches.
3. [tex]\( x = 0.8 \)[/tex]:
[tex]\[ f(0.8) = \frac{3 + 5(0.8)}{2} = \frac{3 + 4}{2} = \frac{7}{2} = 3.5 \][/tex]
The output matches.
4. [tex]\( x = 4.4 \)[/tex]:
[tex]\[ f(4.4) = \frac{3 + 5(4.4)}{2} = \frac{3 + 22}{2} = \frac{25}{2} = 12.5 \][/tex]
The output matches.
Table 3:
[tex]\[ \begin{tabular}{|l|r|r|r|r|} \hline Input ( $x$ ) & 1.1 & 2.4 & 5.3 & 6.9 \\ \hline Output $f(x)$ & 4.25 & 7.5 & 11.75 & 17.25 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = 1.1 \)[/tex]:
[tex]\[ f(1.1) = \frac{3 + 5(1.1)}{2} = \frac{3 + 5.5}{2} = \frac{8.5}{2} = 4.25 \][/tex]
The output matches.
2. [tex]\( x = 2.4 \)[/tex]:
[tex]\[ f(2.4) = \frac{3 + 5(2.4)}{2} = \frac{3 + 12}{2} = \frac{15}{2} = 7.5 \][/tex]
The output matches.
3. [tex]\( x = 5.3 \)[/tex]:
[tex]\[ f(5.3) = \frac{3 + 5(5.3)}{2} = \frac{3 + 26.5}{2} = \frac{29.5}{2} = 14.75 \][/tex]
Given output is 11.75, which is incorrect.
Table 4:
[tex]\[ \begin{tabular}{|l|r|r|r|r|} \hline Input $(x)$ & -2.1 & 0.9 & 1.7 & 3.3 \\ \hline Output $f(x)$ & -3.75 & 3.75 & 5.75 & 9.75 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = -2.1 \)[/tex]:
[tex]\[ f(-2.1) = \frac{3 + 5(-2.1)}{2} = \frac{3 - 10.5}{2} = \frac{-7.5}{2} = -3.75 \][/tex]
The output matches.
2. [tex]\( x = 0.9 \)[/tex]:
[tex]\[ f(0.9) = \frac{3 + 5(0.9)}{2} = \frac{3 + 4.5}{2} = \frac{7.5}{2} = 3.75 \][/tex]
The output matches.
3. [tex]\( x = 1.7 \)[/tex]:
[tex]\[ f(1.7) = \frac{3 + 5(1.7)}{2} = \frac{3 + 8.5}{2} = \frac{11.5}{2} = 5.75 \][/tex]
The output matches.
4. [tex]\( x = 3.3 \)[/tex]:
[tex]\[ f(3.3) = \frac{3 + 5(3.3)}{2} = \frac{3 + 16.5}{2} = \frac{19.5}{2} = 9.75 \][/tex]
The output matches.
Table 5:
[tex]\[ \begin{tabular}{|l|r|r|r|r|} \hline Input $(x)$ & 2.4 & 3.5 & 4.6 & 5.0 \\ \hline Output $f(x)$ & 7.5 & 10.5 & 12.5 & 14 \\ \hline \end{tabular} \][/tex]
1. [tex]\( x = 2.4 \)[/tex]:
[tex]\[ f(2.4) = \frac{3 + 5(2.4)}{2} = \frac{3 + 12}{2} = \frac{15}{2} = 7.5 \][/tex]
The output matches.
2. [tex]\( x = 3.5 \)[/tex]:
[tex]\[ f(3.5) = \frac{3 + 5(3.5)}{2} = \frac{3 + 17.5}{2} = \frac{20.5}{2} = 10.5 \][/tex]
The output matches.
3. [tex]\( x = 4.6 \)[/tex]:
[tex]\[ f(4.6) = \frac{3 + 5(4.6)}{2} = \frac{3 + 23}{2} = \frac{26}{2} = 13 \][/tex]
Given output is 12.5, which is incorrect.
4. [tex]\( x = 5.0 \)[/tex]:
[tex]\[ f(5.0) = \frac{3 + 5(5.0)}{2} = \frac{3 + 25}{2} = \frac{28}{2} = 14 \][/tex]
Output is 14, which is correct.
Therefore, the tables that contain correct pairs of input and output values are Table 2 and Table 4.
