Let's carefully calculate each value of the function [tex]\( f(x) = 4\left(\frac{3}{2}\right)^x \)[/tex] for the given domain [tex]\( \{-3, -2, -1, 0, 1, 2\} \)[/tex]. Then, we will complete the table and proceed to plot the graph.
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[
f(-3) = 4\left(\frac{3}{2}\right)^{-3} = 4 \left(\frac{2}{3}\right)^3 = 4 \times \frac{8}{27} = \frac{32}{27} \approx 1.185
\][/tex]
2. For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 4\left(\frac{3}{2}\right)^{-2} = 4 \left(\frac{2}{3}\right)^2 = 4 \times \frac{4}{9} = \frac{16}{9} \approx 1.778
\][/tex]
3. For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 4\left(\frac{3}{2}\right)^{-1} = 4 \times \frac{2}{3} = \frac{8}{3} \approx 2.667
\][/tex]
4. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 4\left(\frac{3}{2}\right)^0 = 4 \times 1 = 4.0
\][/tex]
5. For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 4\left(\frac{3}{2}\right)^1 = 4 \times \frac{3}{2} = 6.0
\][/tex]
6. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 4\left(\frac{3}{2}\right)^2 = 4 \left(\frac{9}{4}\right) = 9.0
\][/tex]
Thus, the completed table is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) = 4\left(\frac{3}{2}\right)^x \\
\hline
-3 & 1.185 \\
-2 & 1.778 \\
-1 & 2.667 \\
0 & 4.0 \\
1 & 6.0 \\
2 & 9.0 \\
\hline
\end{array}
\][/tex]
To graph the function, you can plot the points [tex]\((-3, 1.185), (-2, 1.778), (-1, 2.667), (0, 4.0), (1, 6.0), (2, 9.0)\)[/tex] on a coordinate plane. The graph will show an increasing exponential function.
