To solve this problem, we need to determine the values of the function [tex]\( g(x) = 6 \left(\frac{3}{2}\right)^x \)[/tex] for the given [tex]\( x \)[/tex]-values: -1, 0, 1, and 2.
Let's evaluate the function for each of these [tex]\( x \)[/tex]-values step-by-step:
1. For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = 6 \left(\frac{3}{2}\right)^{-1}
\][/tex]
Using the properties of exponents, we know that [tex]\( \left(\frac{3}{2}\right)^{-1} = \frac{2}{3} \)[/tex]. Therefore:
[tex]\[
g(-1) = 6 \cdot \frac{2}{3} = 4
\][/tex]
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 6 \left(\frac{3}{2}\right)^0
\][/tex]
Any number raised to the power of 0 is 1. Therefore:
[tex]\[
g(0) = 6 \cdot 1 = 6
\][/tex]
3. For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 6 \left(\frac{3}{2}\right)^1
\][/tex]
Therefore:
[tex]\[
g(1) = 6 \cdot \frac{3}{2} = 9
\][/tex]
4. For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = 6 \left(\frac{3}{2}\right)^2
\][/tex]
First, calculate [tex]\( \left(\frac{3}{2}\right)^2 = \frac{9}{4} \)[/tex]. Therefore:
[tex]\[
g(2) = 6 \cdot \frac{9}{4} = 13.5
\][/tex]
Having calculated these values, we can now fill in the table:
[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, you can plot these points [tex]\((-1, 4)\)[/tex], [tex]\((0, 6)\)[/tex], [tex]\((1, 9)\)[/tex], and [tex]\((2, 13.5)\)[/tex] on the provided grid to visualize the graph of the function [tex]\( g(x) \)[/tex].
