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 calculate the cube root of each given [tex]\( x \)[/tex] value. The [tex]$t$[/tex]-table provides specific [tex]\( x \)[/tex] values, and we will find the corresponding [tex]\( y \)[/tex] values, constructing it step-by-step.
Given:
[tex]\( x = -8, -1, 0, 1, 8 \)[/tex]
[tex]\( y = \sqrt[3]{x} \)[/tex]
1. For [tex]\( x = -8 \)[/tex]:
We find [tex]\( y = \sqrt[3]{-8} \)[/tex].
The cube root of [tex]\( -8 \)[/tex] yields [tex]\( (1.0000000000000002+1.7320508075688772j) \)[/tex].
2. For [tex]\( x = -1 \)[/tex]:
We find [tex]\( y = \sqrt[3]{-1} \)[/tex].
The cube root of [tex]\( -1 \)[/tex] yields [tex]\( (0.5000000000000001+0.8660254037844386j) \)[/tex].
3. For [tex]\( x = 0 \)[/tex]:
We find [tex]\( y = \sqrt[3]{0} \)[/tex].
The cube root of [tex]\( 0 \)[/tex] is simply [tex]\( 0 \)[/tex].
4. For [tex]\( x = 1 \)[/tex]:
We find [tex]\( y = \sqrt[3]{1} \)[/tex].
The cube root of [tex]\( 1 \)[/tex] is simply [tex]\( 1 \)[/tex].
5. For [tex]\( x = 8 \)[/tex]:
We find [tex]\( y = \sqrt[3]{8} \)[/tex].
The cube root of [tex]\( 8 \)[/tex] is simply [tex]\( 2 \)[/tex].
Thus, the completed [tex]\( t \)[/tex]-table looks like this:
[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]
This table gives the precise [tex]\( y \)[/tex]-values for each corresponding [tex]\( x \)[/tex]-value for the function [tex]\( y = \sqrt[3]{x} \)[/tex].
