Fill in the values of [tex]\(Y\)[/tex] for each corresponding [tex]\(X\)[/tex] in the table:

[tex]\[
\begin{tabular}{l|l}
$X$ & $Y = x^2 - 2X - 2$ \\
\hline
1 & \\
2 & \\
3 & \\
0 & \\
-1 & \\
-2 &
\end{tabular}
\][/tex]



Answer :

Alright, let's fill in the table by calculating the values of [tex]\( Y \)[/tex] for each given [tex]\( X \)[/tex] using the function [tex]\( Y = x^2 - 2x - 2 \)[/tex].

1. For [tex]\( X = 1 \)[/tex]:
[tex]\[ Y = 1^2 - 2 \cdot 1 - 2 \\ = 1 - 2 - 2 \\ = -3 \][/tex]

2. For [tex]\( X = 2 \)[/tex]:
[tex]\[ Y = 2^2 - 2 \cdot 2 - 2 \\ = 4 - 4 - 2 \\ = -2 \][/tex]

3. For [tex]\( X = 3 \)[/tex]:
[tex]\[ Y = 3^2 - 2 \cdot 3 - 2 \\ = 9 - 6 - 2 \\ = 1 \][/tex]

4. For [tex]\( X = 0 \)[/tex]:
[tex]\[ Y = 0^2 - 2 \cdot 0 - 2 \\ = 0 - 0 - 2 \\ = -2 \][/tex]

5. For [tex]\( X = -1 \)[/tex]:
[tex]\[ Y = (-1)^2 - 2 \cdot (-1) - 2 \\ = 1 + 2 - 2 \\ = 1 \][/tex]

6. For [tex]\( X = -2 \)[/tex]:
[tex]\[ Y = (-2)^2 - 2 \cdot (-2) - 2 \\ = 4 + 4 - 2 \\ = 6 \][/tex]

Now let's update the table with the calculated values of [tex]\( Y \)[/tex]:

[tex]\[ \begin{tabular}{l|l} $X$ & $Y = x^2 - 2X - 2$ \\ \hline 1 & -3 \\ 2 & -2 \\ 3 & 1 \\ 0 & -2 \\ -1 & 1 \\ -2 & 6 \\ \end{tabular} \][/tex]

So, the completed table is as follows:
[tex]\[ \begin{array}{c|c} X & Y = x^2 - 2X - 2 \\ \hline 1 & -3 \\ 2 & -2 \\ 3 & 1 \\ 0 & -2 \\ -1 & 1 \\ -2 & 6 \\ \end{array} \][/tex]