Let's work through the calculations step-by-step to find the values for [tex]\( d \)[/tex], [tex]\( e \)[/tex], and [tex]\( f \)[/tex]:
1. First, let's calculate [tex]\( d \)[/tex]:
For [tex]\( x = 0 \)[/tex],
[tex]\[ d = \left( \frac{2}{3} \right)^0 \][/tex]
We know that any number raised to the power of 0 is 1. Therefore,
[tex]\[ d = 1.0 \][/tex]
2. Next, let's calculate [tex]\( e \)[/tex]:
For [tex]\( x = 2 \)[/tex],
[tex]\[ e = \left( \frac{2}{3} \right)^2 \][/tex]
When we square [tex]\(\frac{2}{3}\)[/tex], we get:
[tex]\[ e = \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) = \frac{4}{9} = 0.4444444444444444 \][/tex]
So,
[tex]\[ e = 0.4444444444444444 \][/tex]
3. Finally, let's calculate [tex]\( f \)[/tex]:
For [tex]\( x = 4 \)[/tex],
[tex]\[ f = \left( \frac{2}{3} \right)^4 \][/tex]
When we raise [tex]\(\frac{2}{3}\)[/tex] to the 4th power, we perform the multiplication:
[tex]\[ f = \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) = \frac{16}{81} = 0.19753086419753083 \][/tex]
So,
[tex]\[ f = 0.19753086419753083 \][/tex]
Now, let's fill in the values in the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $\left(\frac{2}{3}\right)^x$ \\
\hline
-1 & $\frac{3}{2}$ \\
\hline
0 & 1.0 \\
\hline
2 & 0.4444444444444444 \\
\hline
4 & 0.19753086419753083 \\
\hline
\end{tabular}
\][/tex]
Therefore,
[tex]\[ d = 1.0 \][/tex]
[tex]\[ e = 0.4444444444444444 \][/tex]
[tex]\[ f = 0.19753086419753083 \][/tex]
That concludes the step-by-step solution!
