Answered

Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

A piecewise function [tex]$f(x)$[/tex] is defined as shown.

[tex]\[ f(x)=\left\{\begin{array}{ll}
-\frac{5}{4} x+90, & 0 \leq x\ \textless \ 40 \\
-\frac{3}{8} x+75, & 40 \leq x \leq 200
\end{array}\right. \][/tex]

Which table could be used to graph a piece of the function?

[tex]\[
\begin{tabular}{|c|c|}
\hline $x$ & $y$ \\
\hline 0 & 90 \\
\hline 16 & 85 \\
\hline 40 & 75 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|}
\hline $x$ & $y$ \\
\hline 0 & 90 \\
\hline 40 & 40 \\
\hline 200 & 0 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|}
\hline $x$ & $y$ \\
\hline 40 & 75 \\
\hline 120 & 30 \\
\hline 200 & 0 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|}
\hline $x$ & $y$ \\
\hline 40 & 60 \\
\hline 160 & 15 \\
\hline 200 & 0 \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnswer
Let's analyze the piecewise function [tex]\( f(x) \)[/tex] and calculate the corresponding [tex]\( y \)[/tex]-values for each given [tex]\( x \)[/tex]-value in the tables provided:

The piecewise function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[ f(x) = \left\{ \begin{array}{ll} -\frac{5}{4} x + 90, & \text{for } 0 \leq x < 40 \\ -\frac{3}{8} x + 75, & \text{for } 40 \leq x \leq 200 \end{array} \right. \][/tex]

### 1. First Table
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 0 & 90 \\ \hline 16 & 85 \\ \hline 40 & 75 \\ \hline \end{tabular} \][/tex]

Calculations:

- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\frac{5}{4}(0) + 90 = 90 \][/tex]
- For [tex]\( x = 16 \)[/tex]:
[tex]\[ f(16) = -\frac{5}{4}(16) + 90 = -20 + 90 = 70 \quad \text{(This doesn't match the given table value of 85)} \][/tex]

Since [tex]\( x = 16 \)[/tex] yields 70, not 85, this table is incorrect.

### 2. Second Table
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 0 & 90 \\ \hline 40 & 40 \\ \hline 200 & 0 \\ \hline \end{tabular} \][/tex]

Calculations:

- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\frac{5}{4}(0) + 90 = 90 \][/tex]
- For [tex]\( x = 40 \)[/tex]:
[tex]\[ f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \quad \text{(This doesn't match the given table value of 40)} \][/tex]

Since [tex]\( x = 40 \)[/tex] yields 60, not 40, this table is incorrect.

### 3. Third Table
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 40 & 75 \\ \hline 120 & 30 \\ \hline 200 & 0 \\ \hline \end{tabular} \][/tex]

Calculations:

- For [tex]\( x = 40 \)[/tex]:
[tex]\[ f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \quad \text{(This doesn't match the given table value of 75)} \][/tex]

Since [tex]\( x = 40 \)[/tex] yields 60, not 75, this table is also incorrect.

### 4. Fourth Table
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 40 & 60 \\ \hline 160 & 15 \\ \hline 200 & 0 \\ \hline \end{tabular} \][/tex]

Calculations:

- For [tex]\( x = 40 \)[/tex]:
[tex]\[ f(40) = -\frac{3}{8}(40) + 75 = -15 + 75 = 60 \][/tex]
- For [tex]\( x = 160 \)[/tex]:
[tex]\[ f(160) = -\frac{3}{8}(160) + 75 = -60 + 75 = 15 \][/tex]
- For [tex]\( x = 200 \)[/tex]:
[tex]\[ f(200) = -\frac{3}{8}(200) + 75 = -75 + 75 = 0 \][/tex]

Based on these calculations, the fourth table correctly reflects the piecewise function values:

[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 40 & 60 \\ \hline 160 & 15 \\ \hline 200 & 0 \\ \hline \end{tabular} \][/tex]

Thus, the correct table to use is the fourth one.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.