Let's analyze the functions [tex]\( f(x) = 2^x \)[/tex] and [tex]\( g(x) = \left( \frac{1}{2} \right)^x \)[/tex] by looking at the values given in the table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) = 2^x & g(x) = \left( \frac{1}{2} \right)^x \\
\hline
2 & 4 & \frac{1}{4} \\
\hline
1 & 2 & \frac{1}{2} \\
\hline
0 & 1 & 1 \\
\hline
-1 & \frac{1}{2} & 2 \\
\hline
-2 & \frac{1}{4} & 4 \\
\hline
\end{array}
\][/tex]
Now, let's compare the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
1. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2^2 = 4, \quad g(2) = \left( \frac{1}{2} \right)^2 = \frac{1}{4}
\][/tex]
2. For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2^1 = 2, \quad g(1) = \left( \frac{1}{2} \right)^1 = \frac{1}{2}
\][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2^0 = 1, \quad g(0) = \left( \frac{1}{2} \right)^0 = 1
\][/tex]
4. For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 2^{-1} = \frac{1}{2}, \quad g(-1) = \left( \frac{1}{2} \right)^{-1} = 2
\][/tex]
5. For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 2^{-2} = \frac{1}{4}, \quad g(-2) = \left( \frac{1}{2} \right)^{-2} = 4
\][/tex]
Notice that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are related in such a way that [tex]\( f(x) = g(-x) \)[/tex]. This indicates that the values of [tex]\( g(x) \)[/tex] are essentially the values of [tex]\( f(x) \)[/tex] when the signs of [tex]\( x \)[/tex] are flipped. This relationship shows that the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are reflections of each other over the [tex]\( y \)[/tex]-axis.
Therefore, the correct conclusion about [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] from the table is:
- The functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are reflections over the [tex]\( y \)[/tex]-axis.
