Answer :
To identify the table that represents the second piece of the function \( f(x) \), defined as:
[tex]\[ f(x) = \left\{ \begin{array}{ll} -3.5x + 0.5, & \text{ for } x < 1 \\ 8 - 2x, & \text{ for } x \geq 1 \end{array} \right. \][/tex]
we will evaluate the second part of this piecewise function for \( x \geq 1 \), at several integer values of \( x \).
### Evaluating \( f(x) = 8 - 2x \):
1. For \( x = 1 \):
[tex]\[ f(1) = 8 - 2(1) = 8 - 2 = 6 \][/tex]
2. For \( x = 2 \):
[tex]\[ f(2) = 8 - 2(2) = 8 - 4 = 4 \][/tex]
3. For \( x = 3 \):
[tex]\[ f(3) = 8 - 2(3) = 8 - 6 = 2 \][/tex]
Now that we have evaluated the function, we can compile the results into a table:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 1 & 6 \\ \hline 2 & 4 \\ \hline 3 & 2 \\ \hline \end{array} \][/tex]
Hence, the table representing the second piece of the function \( f(x) = 8 - 2x \) for \( x \geq 1 \) is:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 1 & 6 \\ \hline 2 & 4 \\ \hline 3 & 2 \\ \hline \end{array} \][/tex]
This corresponds to the third table provided in the question.
[tex]\[ f(x) = \left\{ \begin{array}{ll} -3.5x + 0.5, & \text{ for } x < 1 \\ 8 - 2x, & \text{ for } x \geq 1 \end{array} \right. \][/tex]
we will evaluate the second part of this piecewise function for \( x \geq 1 \), at several integer values of \( x \).
### Evaluating \( f(x) = 8 - 2x \):
1. For \( x = 1 \):
[tex]\[ f(1) = 8 - 2(1) = 8 - 2 = 6 \][/tex]
2. For \( x = 2 \):
[tex]\[ f(2) = 8 - 2(2) = 8 - 4 = 4 \][/tex]
3. For \( x = 3 \):
[tex]\[ f(3) = 8 - 2(3) = 8 - 6 = 2 \][/tex]
Now that we have evaluated the function, we can compile the results into a table:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 1 & 6 \\ \hline 2 & 4 \\ \hline 3 & 2 \\ \hline \end{array} \][/tex]
Hence, the table representing the second piece of the function \( f(x) = 8 - 2x \) for \( x \geq 1 \) is:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 1 & 6 \\ \hline 2 & 4 \\ \hline 3 & 2 \\ \hline \end{array} \][/tex]
This corresponds to the third table provided in the question.