Answer :
To determine which table could be used to graph the piecewise function [tex]\( f(x) \)[/tex], we will evaluate each [tex]\( x \)[/tex]-value in the tables using the function definitions and see if the corresponding [tex]\( y \)[/tex]-values match.
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[ f(x)=\left\{\begin{array}{ll} -\frac{5}{4} x+90, & 0 \leq x<40 \\ -\frac{3}{8} x+75, & 40 \leq x \leq 200 \end{array}\right. \][/tex]
Let's check the values in each table one by one.
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 90 \\ \hline 16 & 85 \\ \hline 40 & 75 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = -\frac{5}{4}(0) + 90 = 90 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 90.
- For [tex]\( x = 16 \)[/tex]: [tex]\( f(16) = -\frac{5}{4}(16) + 90 = -20 + 90 = 70 \)[/tex]. This does not match the [tex]\( y \)[/tex]-value 85.
- For [tex]\( x = 40 \)[/tex]: [tex]\( f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \)[/tex]. This does not match the [tex]\( y \)[/tex]-value 75.
Since not all [tex]\( y \)[/tex]-values match, Table 1 is not correct.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 90 \\ \hline 40 & 40 \\ \hline 200 & 0 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = -\frac{5}{4}(0) + 90 = 90 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 90.
- For [tex]\( x = 40 \)[/tex]: [tex]\( f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \)[/tex]. This does not match the [tex]\( y \)[/tex]-value 40.
- For [tex]\( x = 200 \)[/tex]: [tex]\( f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 0.
Since not all [tex]\( y \)[/tex]-values match, Table 2 is not correct.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 40 & 75 \\ \hline 120 & 30 \\ \hline 200 & 0 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 40 \)[/tex]: [tex]\( f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \)[/tex]. This does not match the [tex]\( y \)[/tex]-value 75.
- For [tex]\( x = 120 \)[/tex]: [tex]\( f(120) = -\frac{3}{8}(120) + 75 = -45 + 75 = 30 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 30.
- For [tex]\( x = 200 \)[/tex]: [tex]\( f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 0.
Since not all [tex]\( y \)[/tex]-values match, Table 3 is not correct.
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 40 & 60 \\ \hline 160 & 15 \\ \hline 200 & 0 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 40 \)[/tex]: [tex]\( f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 60.
- For [tex]\( x = 160 \)[/tex]: [tex]\( f(160) = -\frac{3}{8}(160) + 75 = -60 + 75 = 15 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 15.
- For [tex]\( x = 200 \)[/tex]: [tex]\( f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 0.
Since all [tex]\( y \)[/tex]-values match, Table 4 is correct.
Therefore, the table that could be used to graph a piece of the function is:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 40 & 60 \\ \hline 160 & 15 \\ \hline 200 & 0 \\ \hline \end{array} \][/tex]
So, the correct table is Table 4.
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[ f(x)=\left\{\begin{array}{ll} -\frac{5}{4} x+90, & 0 \leq x<40 \\ -\frac{3}{8} x+75, & 40 \leq x \leq 200 \end{array}\right. \][/tex]
Let's check the values in each table one by one.
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 90 \\ \hline 16 & 85 \\ \hline 40 & 75 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = -\frac{5}{4}(0) + 90 = 90 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 90.
- For [tex]\( x = 16 \)[/tex]: [tex]\( f(16) = -\frac{5}{4}(16) + 90 = -20 + 90 = 70 \)[/tex]. This does not match the [tex]\( y \)[/tex]-value 85.
- For [tex]\( x = 40 \)[/tex]: [tex]\( f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \)[/tex]. This does not match the [tex]\( y \)[/tex]-value 75.
Since not all [tex]\( y \)[/tex]-values match, Table 1 is not correct.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 90 \\ \hline 40 & 40 \\ \hline 200 & 0 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = -\frac{5}{4}(0) + 90 = 90 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 90.
- For [tex]\( x = 40 \)[/tex]: [tex]\( f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \)[/tex]. This does not match the [tex]\( y \)[/tex]-value 40.
- For [tex]\( x = 200 \)[/tex]: [tex]\( f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 0.
Since not all [tex]\( y \)[/tex]-values match, Table 2 is not correct.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 40 & 75 \\ \hline 120 & 30 \\ \hline 200 & 0 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 40 \)[/tex]: [tex]\( f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \)[/tex]. This does not match the [tex]\( y \)[/tex]-value 75.
- For [tex]\( x = 120 \)[/tex]: [tex]\( f(120) = -\frac{3}{8}(120) + 75 = -45 + 75 = 30 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 30.
- For [tex]\( x = 200 \)[/tex]: [tex]\( f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 0.
Since not all [tex]\( y \)[/tex]-values match, Table 3 is not correct.
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 40 & 60 \\ \hline 160 & 15 \\ \hline 200 & 0 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 40 \)[/tex]: [tex]\( f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 60.
- For [tex]\( x = 160 \)[/tex]: [tex]\( f(160) = -\frac{3}{8}(160) + 75 = -60 + 75 = 15 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 15.
- For [tex]\( x = 200 \)[/tex]: [tex]\( f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \)[/tex]. This matches the [tex]\( y \)[/tex]-value 0.
Since all [tex]\( y \)[/tex]-values match, Table 4 is correct.
Therefore, the table that could be used to graph a piece of the function is:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 40 & 60 \\ \hline 160 & 15 \\ \hline 200 & 0 \\ \hline \end{array} \][/tex]
So, the correct table is Table 4.
