To complete the data table for the function [tex]\( f(x) = \sqrt{-x-2} + 2 \)[/tex], we need to calculate the value of [tex]\( f(x) \)[/tex] for each given [tex]\( x \)[/tex] value.
Let's evaluate the function step by step for each [tex]\( x \)[/tex]:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = \sqrt{-(-2)-2} + 2 = \sqrt{2-2} + 2 = \sqrt{0} + 2 = 0 + 2 = 2.0
\][/tex]
2. For [tex]\( x = -3 \)[/tex]:
[tex]\[
f(-3) = \sqrt{-(-3)-2} + 2 = \sqrt{3-2} + 2 = \sqrt{1} + 2 = 1 + 2 = 3.0
\][/tex]
3. For [tex]\( x = -6 \)[/tex]:
[tex]\[
f(-6) = \sqrt{-(-6)-2} + 2 = \sqrt{6-2} + 2 = \sqrt{4} + 2 = 2 + 2 = 4.0
\][/tex]
4. For [tex]\( x = -11 \)[/tex]:
[tex]\[
f(-11) = \sqrt{-(-11)-2} + 2 = \sqrt{11-2} + 2 = \sqrt{9} + 2 = 3 + 2 = 5.0
\][/tex]
Now, let's fill in the data table with the calculated values of [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{c|cccc}
X & -2 & -3 & -6 & -11 \\
\hline
Y & 2.0 & 3.0 & 4.0 & 5.0 \\
\end{array}
\][/tex]
So, the completed data table is:
[tex]\[
\begin{array}{c|cccc}
X & -2 & -3 & -6 & -11 \\
\hline
Y & 2.0 & 3.0 & 4.0 & 5.0 \\
\end{array}
\][/tex]
