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Sagot :
To determine which of the given tables represents a non-linear function, we need to check the consistency of the differences in the [tex]\(y\)[/tex] values when the [tex]\(x\)[/tex] values increase by a constant amount. A linear function will exhibit a constant rate of change, i.e., the difference between consecutive [tex]\(y\)[/tex] values will be the same.
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 17 & 20 \\ \hline 18 & 19 \\ \hline 19 & 18 \\ \hline 20 & 17 \\ \hline \end{array} \][/tex]
Differences in [tex]\(y\)[/tex] values:
[tex]\[ \begin{align*} 19 - 20 &= -1 \\ 18 - 19 &= -1 \\ 17 - 18 &= -1 \end{align*} \][/tex]
All the differences are consistent [tex]\((-1)\)[/tex], indicating this table represents a linear function.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 17 & -16 \\ \hline 18 & -17 \\ \hline 19 & -18 \\ \hline 20 & -19 \\ \hline \end{array} \][/tex]
Differences in [tex]\(y\)[/tex] values:
[tex]\[ \begin{align*} -17 - (-16) &= -1 \\ -18 - (-17) &= -1 \\ -19 - (-18) &= -1 \end{align*} \][/tex]
All the differences are consistent [tex]\((-1)\)[/tex], indicating this table represents a linear function.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 17 & 16 \\ \hline 18 & 17 \\ \hline 19 & 19 \\ \hline 20 & 20 \\ \hline \end{array} \][/tex]
Differences in [tex]\(y\)[/tex] values:
[tex]\[ \begin{align*} 17 - 16 &= 1 \\ 19 - 17 &= 2 \\ 20 - 19 &= 1 \end{align*} \][/tex]
The differences [tex]\( (1, 2, 1) \)[/tex] are not consistent, indicating that this table does not represent a linear function.
### Conclusion:
By examining the differences in [tex]\(y\)[/tex] values, we determine that Table 3 does not represent a linear function. Therefore, the table representing a non-linear function is Table 3.
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 17 & 20 \\ \hline 18 & 19 \\ \hline 19 & 18 \\ \hline 20 & 17 \\ \hline \end{array} \][/tex]
Differences in [tex]\(y\)[/tex] values:
[tex]\[ \begin{align*} 19 - 20 &= -1 \\ 18 - 19 &= -1 \\ 17 - 18 &= -1 \end{align*} \][/tex]
All the differences are consistent [tex]\((-1)\)[/tex], indicating this table represents a linear function.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 17 & -16 \\ \hline 18 & -17 \\ \hline 19 & -18 \\ \hline 20 & -19 \\ \hline \end{array} \][/tex]
Differences in [tex]\(y\)[/tex] values:
[tex]\[ \begin{align*} -17 - (-16) &= -1 \\ -18 - (-17) &= -1 \\ -19 - (-18) &= -1 \end{align*} \][/tex]
All the differences are consistent [tex]\((-1)\)[/tex], indicating this table represents a linear function.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 17 & 16 \\ \hline 18 & 17 \\ \hline 19 & 19 \\ \hline 20 & 20 \\ \hline \end{array} \][/tex]
Differences in [tex]\(y\)[/tex] values:
[tex]\[ \begin{align*} 17 - 16 &= 1 \\ 19 - 17 &= 2 \\ 20 - 19 &= 1 \end{align*} \][/tex]
The differences [tex]\( (1, 2, 1) \)[/tex] are not consistent, indicating that this table does not represent a linear function.
### Conclusion:
By examining the differences in [tex]\(y\)[/tex] values, we determine that Table 3 does not represent a linear function. Therefore, the table representing a non-linear function is Table 3.
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