Sure, let's evaluate the function [tex]\( f(x) = -2 \cdot 3^x \)[/tex] for each of the given [tex]\( x \)[/tex] values: -2, -1, 0, 1, and 2.
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = -2 \cdot 3^{-2}
\][/tex]
Recall that [tex]\( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \)[/tex]:
[tex]\[
f(-2) = -2 \cdot \frac{1}{9} = -\frac{2}{9} \approx -0.2222
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = -2 \cdot 3^{-1}
\][/tex]
Recall that [tex]\( 3^{-1} = \frac{1}{3} \)[/tex]:
[tex]\[
f(-1) = -2 \cdot \frac{1}{3} = -\frac{2}{3} \approx -0.6667
\][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -2 \cdot 3^0
\][/tex]
Recall that [tex]\( 3^0 = 1 \)[/tex]:
[tex]\[
f(0) = -2 \cdot 1 = -2
\][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = -2 \cdot 3^1
\][/tex]
Recall that [tex]\( 3^1 = 3 \)[/tex]:
[tex]\[
f(1) = -2 \cdot 3 = -6
\][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = -2 \cdot 3^2
\][/tex]
Recall that [tex]\( 3^2 = 9 \)[/tex]:
[tex]\[
f(2) = -2 \cdot 9 = -18
\][/tex]
With these evaluations, we can now fill in the table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & -2 & -1 & 0 & 1 & 2 \\
\hline
$f(x)$ & -0.2222 & -0.6667 & -2 & -6 & -18 \\
\hline
\end{tabular}
\][/tex]
