Let's determine the values for [tex]\( d \)[/tex], [tex]\( e \)[/tex], and [tex]\( f \)[/tex] by evaluating the given function [tex]\( \left(\frac{2}{3}\right)^x \)[/tex] for the specific values of [tex]\( x \)[/tex].
1. Step 1: Evaluate at [tex]\( x = 0 \)[/tex]
The general rule for any non-zero number raised to the power of 0 is that it equals 1:
[tex]\[
\left(\frac{2}{3}\right)^0 = 1
\][/tex]
Therefore, [tex]\( d = 1 \)[/tex].
2. Step 2: Evaluate at [tex]\( x = 2 \)[/tex]
[tex]\[
\left(\frac{2}{3}\right)^2 = \left(\frac{2}{3} \times \frac{2}{3}\right) = \frac{4}{9} \approx 0.4444444444444444
\][/tex]
Therefore, [tex]\( e \approx 0.4444444444444444 \)[/tex].
3. Step 3: Evaluate at [tex]\( x = 4 \)[/tex]
[tex]\[
\left(\frac{2}{3}\right)^4 = \left(\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}\right) = \frac{16}{81} \approx 0.19753086419753083
\][/tex]
Therefore, [tex]\( f \approx 0.19753086419753083 \)[/tex].
Let's summarize the results into the table and the final answers for [tex]\( d \)[/tex], [tex]\( e \)[/tex], and [tex]\( f \)[/tex]:
\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$\left(\frac{2}{3}\right)^x$[/tex] \\
\hline -1 & [tex]$\frac{3}{2}$[/tex] \\
\hline 0 & [tex]\( d = 1 \)[/tex] \\
\hline 2 & [tex]\( e \approx 0.4444444444444444 \)[/tex] \\
\hline 4 & [tex]\( f \approx 0.19753086419753083 \)[/tex] \\
\hline
\end{tabular}
Thus, the values are:
[tex]\[
d = 1, \quad e \approx 0.4444444444444444, \quad f \approx 0.19753086419753083
\][/tex]
