To determine which table correctly represents the function values for the piecewise function [tex]\( f(x) \)[/tex], we need to evaluate [tex]\( f(x) \)[/tex] at the given [tex]\( x \)[/tex] values from each table and compare the results.
The piecewise function is provided 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 verify each table step-by-step:
### 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]\[
f(0) = -\frac{5}{4}(0) + 90 = 90 \quad (\text{matches})
\][/tex]
- For [tex]\( x = 16 \)[/tex]:
[tex]\[
f(16) = -\frac{5}{4}(16) + 90 = -20 + 90 = 70 \quad (\text{does not match})
\][/tex]
- For [tex]\( x = 40 \)[/tex]:
[tex]\[
f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \quad (\text{does not match})
\][/tex]
Since [tex]\( f(x) \)[/tex] does not match for all [tex]\( x \)[/tex] values in the table, Table 1 is incorrect.
### 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]\[
f(0) = -\frac{5}{4}(0) + 90 = 90 \quad (\text{matches})
\][/tex]
- For [tex]\( x = 40 \)[/tex]:
[tex]\[
f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \quad (\text{does not match})
\][/tex]
- For [tex]\( x = 200 \)[/tex]:
[tex]\[
f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \quad (\text{matches})
\][/tex]
Since [tex]\( f(x) \)[/tex] does not match for all [tex]\( x \)[/tex] values in the table, Table 2 is incorrect.
### 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]\[
f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \quad (\text{does not match})
\][/tex]
- For [tex]\( x = 120 \)[/tex]:
[tex]\[
f(120) = -\frac{3}{8}(120) + 75 = -45 + 75 = 30 \quad (\text{matches})
\][/tex]
- For [tex]\( x = 200 \)[/tex]:
[tex]\[
f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \quad (\text{matches})
\][/tex]
Since [tex]\( f(x) \)[/tex] does not match for all [tex]\( x \)[/tex] values in the table, Table 3 is incorrect.
### 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]\[
f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \quad (\text{matches})
\][/tex]
- For [tex]\( x = 160 \)[/tex]:
[tex]\[
f(160) = -\frac{3}{8}(160) + 75 = -60 + 75 = 15 \quad (\text{matches})
\][/tex]
- For [tex]\( x = 200 \)[/tex]:
[tex]\[
f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \quad (\text{matches})
\][/tex]
Since [tex]\( f(x) \)[/tex] matches for all [tex]\( x \)[/tex] values in the table, Table 4 is correct.
Thus, the correct table is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
40 & 60 \\
\hline
160 & 15 \\
\hline
200 & 0 \\
\hline
\end{array}
\][/tex]
