To fill in the [tex]\( y \)[/tex] values for the function [tex]\( y = \sqrt[3]{x} \)[/tex], we will evaluate the function at each given [tex]\( x \)[/tex]-value. Here's the process:
1. Calculate [tex]\( y \)[/tex] for [tex]\( x = -8 \)[/tex]:
- [tex]\( y = \sqrt[3]{-8} \)[/tex].
- The result is [tex]\( y = (1.0000000000000002 + 1.7320508075688772i) \)[/tex]. This value is a complex number.
2. Calculate [tex]\( y \)[/tex] for [tex]\( x = -1 \)[/tex]:
- [tex]\( y = \sqrt[3]{-1} \)[/tex].
- The result is [tex]\( y = (0.5000000000000001 + 0.8660254037844386i) \)[/tex]. This is also a complex number.
3. Calculate [tex]\( y \)[/tex] for [tex]\( x = 0 \)[/tex]:
- [tex]\( y = \sqrt[3]{0} \)[/tex].
- The result is [tex]\( y = 0.0 \)[/tex].
4. Calculate [tex]\( y \)[/tex] for [tex]\( x = 1 \)[/tex]:
- [tex]\( y = \sqrt[3]{1} \)[/tex].
- The result is [tex]\( y = 1.0 \)[/tex].
5. Calculate [tex]\( y \)[/tex] for [tex]\( x = 8 \)[/tex]:
- [tex]\( y = \sqrt[3]{8} \)[/tex].
- The result is [tex]\( y = 2.0 \)[/tex].
Now, we can fill in the [tex]\( t \)[/tex]-table with these computed [tex]\( y \)[/tex]-values:
[tex]\[
\begin{tabular}{c|c|}
x & y \\
\hline
-8 & (1.0000000000000002 + 1.7320508075688772i) \\
-1 & (0.5000000000000001 + 0.8660254037844386i) \\
0 & 0.0 \\
1 & 1.0 \\
8 & 2.0 \\
\end{tabular}
\][/tex]
Here is the final [tex]\( t \)[/tex]-table with all the [tex]\( y \)[/tex]-values for the given [tex]\( x \)[/tex]-values:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
-8 & (1.0000000000000002 + 1.7320508075688772i) \\
-1 & (0.5000000000000001 + 0.8660254037844386i) \\
0 & 0.0 \\
1 & 1.0 \\
8 & 2.0 \\
\end{array}
\][/tex]
