To find the values of the function [tex]\( g(x) = \left( \frac{1}{8} \right)^x \)[/tex] for the given [tex]\( x \)[/tex]-values in the table, let's evaluate the function step-by-step for each [tex]\( x \)[/tex].
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) = \left( \frac{1}{8} \right)^{-2}
\][/tex]
By the property of exponents, [tex]\( \left( \frac{1}{a} \right)^{-b} = a^b \)[/tex]:
[tex]\[
\left( \frac{1}{8} \right)^{-2} = 8^2 = 64
\][/tex]
So, [tex]\( g(-2) = 64.0 \)[/tex].
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = \left( \frac{1}{8} \right)^{-1}
\][/tex]
Similarly, by the property of exponents:
[tex]\[
\left( \frac{1}{8} \right)^{-1} = 8
\][/tex]
So, [tex]\( g(-1) = 8.0 \)[/tex].
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = \left( \frac{1}{8} \right)^0
\][/tex]
Any non-zero number raised to the power of 0 is 1:
[tex]\[
\left( \frac{1}{8} \right)^0 = 1
\][/tex]
So, [tex]\( g(0) = 1.0 \)[/tex].
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = \left( \frac{1}{8} \right)^1
\][/tex]
Any number raised to the power of 1 is the number itself:
[tex]\[
\left( \frac{1}{8} \right)^1 = \frac{1}{8}
\][/tex]
So, [tex]\( g(1) = 0.125 \)[/tex].
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = \left( \frac{1}{8} \right)^2
\][/tex]
By the property of exponents:
[tex]\[
\left( \frac{1}{8} \right)^2 = \frac{1}{8^2} = \frac{1}{64}
\][/tex]
So, [tex]\( g(2) = 0.015625 \)[/tex].
Now, we can fill in the table with these values.
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$g(x)$[/tex] \\
\hline
-2 & 64.0 \\
\hline
-1 & 8.0 \\
\hline
0 & 1.0 \\
\hline
1 & 0.125 \\
\hline
2 & 0.015625 \\
\hline
\end{tabular}
