To determine the corresponding values of [tex]\( g(x) \)[/tex] for the given [tex]\( x \)[/tex]-values, we need to evaluate the function [tex]\( g(x) = 6 \left( \frac{3}{2} \right)^x \)[/tex] at each of these [tex]\( x \)[/tex]-values.
1. For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = 6 \left( \frac{3}{2} \right)^{-1} = 6 \left( \frac{2}{3} \right) = 4
\][/tex]
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 6 \left( \frac{3}{2} \right)^0 = 6 \cdot 1 = 6
\][/tex]
3. For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 6 \left( \frac{3}{2} \right)^1 = 6 \cdot \frac{3}{2} = 9
\][/tex]
4. For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = 6 \left( \frac{3}{2} \right)^2 = 6 \cdot \left( \frac{3}{2} \right)^2 = 6 \cdot \frac{9}{4} = 13.5
\][/tex]
Now, we can fill in the table with the corresponding [tex]\( g(x) \)[/tex] values:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
x & -1 & 0 & 1 & 2 \\
\hline
g(x) & 4 & 6 & 9 & 13.5 \\
\hline
\end{tabular}
\][/tex]
Next, we need to plot these points on the graph.
The points to plot are:
- [tex]\( (-1, 4) \)[/tex]
- [tex]\( (0, 6) \)[/tex]
- [tex]\( (1, 9) \)[/tex]
- [tex]\( (2, 13.5) \)[/tex]
These points can be plotted on a [tex]\( (x, y) \)[/tex] coordinate plane. Make sure to label your axis and plot each point accurately.
