Let's consider the function [tex]\( f(x) = 2^x \)[/tex]. We need to evaluate [tex]\( f(x) \)[/tex] at specific [tex]\( x \)[/tex]-values and analyze its behavior.
1. First, we'll calculate [tex]\( f(x) \)[/tex] for each given [tex]\( x \)[/tex]-value:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 2^{-2} = \frac{1}{2^2} = \frac{1}{4} = 0.25
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 2^{-1} = \frac{1}{2} = 0.5
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2^0 = 1
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2^1 = 2
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2^2 = 4
\][/tex]
2. Now, we will fill in the table with the computed values:
[tex]\[
\begin{tabular}{|c|c|cc|c|c|}
\hline
$x$ & -2 & -1 & 0 & 1 & 2 \\
\hline
$f(x)$ & 0.25 & 0.5 & 1 & 2 & 4 \\
\hline
\end{tabular}
\][/tex]
3. Next, we analyze the end behavior of the function.
[tex]\[
\text{As } x \to \infty, f(x) = 2^x \to \infty
\][/tex]
[tex]\[
\text{As } x \to -\infty, f(x) = 2^x \to 0
\][/tex]
Therefore, the end behavior can be summarized as:
[tex]\[
\text{As } x \to \infty, y \to \infty
\][/tex]
[tex]\[
\text{As } x \to -\infty, y \to 0
\][/tex]
So we can complete the end behavior statement as follows:
End Behavior:
[tex]\[
\text{As } x \rightarrow \infty, y \rightarrow \infty \\
\][/tex]
[tex]\[
\text{And as } x \rightarrow -\infty, y \rightarrow 0
\][/tex]
In summary, we have evaluated the function [tex]\( f(x) = 2^x \)[/tex] at the given points, completed the table, and described the function's end behavior.
