Sure, let's work through the solution step-by-step to fill out the table given the function [tex]\( f(x) = 3 \cdot (0.5)^x \)[/tex].
1. Understanding the Function:
The function given is [tex]\( f(x) = 3 \cdot (0.5)^x \)[/tex]. This means for any value of [tex]\( x \)[/tex], you raise [tex]\( 0.5 \)[/tex] to the power of [tex]\( x \)[/tex] and then multiply the result by 3.
2. Calculating [tex]\( f(x) \)[/tex] for Different Values of [tex]\( x \)[/tex]:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 3 \cdot (0.5)^{-2}
\][/tex]
Recall that raised to a negative power, [tex]\( (a^{-n} = \frac{1}{a^n}) \)[/tex]:
[tex]\[
(0.5)^{-2} = \frac{1}{(0.5)^2} = \frac{1}{0.25} = 4
\][/tex]
Therefore,
[tex]\[
f(-2) = 3 \cdot 4 = 12
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 3 \cdot (0.5)^{-1}
\][/tex]
[tex]\[
(0.5)^{-1} = \frac{1}{0.5} = 2
\][/tex]
Therefore,
[tex]\[
f(-1) = 3 \cdot 2 = 6
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3 \cdot (0.5)^0
\][/tex]
Any number raised to the power of 0 is 1:
[tex]\[
(0.5)^0 = 1
\][/tex]
Therefore,
[tex]\[
f(0) = 3 \cdot 1 = 3
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3 \cdot (0.5)^1
\][/tex]
[tex]\[
(0.5)^1 = 0.5
\][/tex]
Therefore,
[tex]\[
f(1) = 3 \cdot 0.5 = 1.5
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 3 \cdot (0.5)^2
\][/tex]
[tex]\[
(0.5)^2 = 0.25
\][/tex]
Therefore,
[tex]\[
f(2) = 3 \cdot 0.25 = 0.75
\][/tex]
3. Filling in the Table:
Now that we have computed [tex]\( f(x) \)[/tex] for each value of [tex]\( x \)[/tex], we can fill in the table:
[tex]\[
\begin{tabular}{|r|r|r|r|r|r|}
\hline
x & -2 & -1 & 0 & 1 & 2 \\
\hline
f(x) & 12 & 6 & 3 & 1.5 & 0.75 \\
\hline
\end{tabular}
\][/tex]
So your completed table should look exactly like this:
[tex]\[
\begin{tabular}{|r|r|r|r|r|r|}
\hline
x & -2 & -1 & 0 & 1 & 2 \\
\hline
f(x) & 12 & 6 & 3 & 1.5 & 0.75 \\
\hline
\end{tabular}
\][/tex]
