To fill in the [tex]$y$[/tex]-values of the [tex]$t$[/tex]-table for the function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to evaluate the cube root for each corresponding [tex]$x$[/tex]-value. Here are the steps for determining each [tex]$y$[/tex]-value:
1. When [tex]\( x = -8 \)[/tex]:
The value of [tex]\( y \)[/tex] is the cube root of [tex]\( -8 \)[/tex].
[tex]\[
y = \sqrt[3]{-8} = (1.0000000000000002 + 1.7320508075688772j)
\][/tex]
2. When [tex]\( x = -1 \)[/tex]:
The value of [tex]\( y \)[/tex] is the cube root of [tex]\( -1 \)[/tex].
[tex]\[
y = \sqrt[3]{-1} = (0.5000000000000001 + 0.8660254037844386j)
\][/tex]
3. When [tex]\( x = 0 \)[/tex]:
The value of [tex]\( y \)[/tex] is the cube root of [tex]\( 0 \)[/tex].
[tex]\[
y = \sqrt[3]{0} = 0.0
\][/tex]
4. When [tex]\( x = 1 \)[/tex]:
The value of [tex]\( y \)[/tex] is the cube root of [tex]\( 1 \)[/tex].
[tex]\[
y = \sqrt[3]{1} = 1.0
\][/tex]
5. When [tex]\( x = 8 \)[/tex]:
The value of [tex]\( y \)[/tex] is the cube root of [tex]\( 8 \)[/tex].
[tex]\[
y = \sqrt[3]{8} = 2.0
\][/tex]
Thus, the completed [tex]\( t \)[/tex]-table is:
[tex]\[
\begin{tabular}{c|c}
$x$ & $y$ \\
\hline
-8 & (1.0000000000000002+1.7320508075688772j) \\
-1 & (0.5000000000000001+0.8660254037844386j) \\
0 & 0.0 \\
1 & 1.0 \\
8 & 2.0 \\
\end{tabular}
\][/tex]
