Let's solve the problem step-by-step using the given function [tex]\( f(x) = (0.5)^x \)[/tex].
### 1. Completing the Table
First, we need to fill out the table by calculating [tex]\( f(x) \)[/tex] for each given value of [tex]\( x \)[/tex].
#### Calculations:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = (0.5)^{-2} = \left(\frac{1}{2}\right)^{-2} = 2^2 = 4
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = (0.5)^{-1} = \left(\frac{1}{2}\right)^{-1} = 2
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = (0.5)^0 = 1
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = (0.5)^1 = 0.5
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = (0.5)^2 = 0.25
\][/tex]
So, the table becomes:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & -2 & -1 & 0 & 1 & 2 \\
$f(x)$ & 4.0 & 2.0 & 1.0 & 0.5 & 0.25 \\
\hline
\end{tabular}
\][/tex]
### 2. Interpreting Constants
The function [tex]\( f(x) = (0.5)^x \)[/tex] can be expressed as [tex]\( f(x) = 1 \cdot \left(\frac{1}{2}\right)^x \)[/tex], where the constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 0.5 \)[/tex]
### 3. End Behavior
- As [tex]\( x \to \infty \)[/tex] (positive infinity):
[tex]\[
\lim_{x \to \infty} (0.5)^x = 0
\][/tex]
So, when [tex]\( x \to \infty \)[/tex], [tex]\( y \to 0 \)[/tex].
- As [tex]\( x \to -\infty \)[/tex] (negative infinity):
[tex]\[
\lim_{x \to -\infty} (0.5)^x = \infty
\][/tex]
So, when [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex].
### Final Answer
[tex]\[
f(x) = (0.5)^x
\][/tex]
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & -2 & -1 & 0 & 1 & 2 \\
$f(x)$ & 4.0 & 2.0 & 1.0 & 0.5 & 0.25 \\
\hline
\end{tabular}
\][/tex]
Hint: [tex]\( f(x) = (0.5)^x \)[/tex], and [tex]\( f(x) = 1 \left(\frac{1}{2}\right)^x \)[/tex]
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 0.5 \)[/tex]
- [tex]\( \text{Interpretation} = 1 \cdot \left(\frac{1}{2}\right)^x \)[/tex]
End Behavior:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to 0 \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex]
Therefore:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow 0 \)[/tex]
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex]
