To determine the relationship between the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] given in the table, we should carefully analyze their values for different [tex]\( x \)[/tex] values. Here's the table for reference:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
$x$ & $f(x)=2^x$ & $g(x)=\left(\frac{1}{2}\right)^x$ \\
\hline
2 & 4 & \frac{1}{4} \\
\hline
1 & 2 & \frac{1}{2} \\
\hline
0 & 1 & 1 \\
\hline
-1 & \frac{1}{2} & 2 \\
\hline
-2 & \frac{1}{4} & 4 \\
\hline
\end{tabular}
\][/tex]
Let's compare the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] at each given [tex]\( x \)[/tex]:
1. When [tex]\( x = 2 \)[/tex]:
- [tex]\( f(2) = 2^2 = 4 \)[/tex]
- [tex]\( g(2) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)[/tex]
2. When [tex]\( x = 1 \)[/tex]:
- [tex]\( f(1) = 2^1 = 2 \)[/tex]
- [tex]\( g(1) = \left(\frac{1}{2}\right)^1 = \frac{1}{2} \)[/tex]
3. When [tex]\( x = 0 \)[/tex]:
- [tex]\( f(0) = 2^0 = 1 \)[/tex]
- [tex]\( g(0) = \left(\frac{1}{2}\right)^0 = 1 \)[/tex]
4. When [tex]\( x = -1 \)[/tex]:
- [tex]\( f(-1) = 2^{-1} = \frac{1}{2} \)[/tex]
- [tex]\( g(-1) = \left(\frac{1}{2}\right)^{-1} = 2 \)[/tex]
5. When [tex]\( x = -2 \)[/tex]:
- [tex]\( f(-2) = 2^{-2} = \frac{1}{4} \)[/tex]
- [tex]\( g(-2) = \left(\frac{1}{2}\right)^{-2} = 4 \)[/tex]
By examining these values, we can notice the following pattern:
- For positive [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] yields values greater than 1, while [tex]\( g(x) \)[/tex] yields values less than 1.
- Both functions yield 1 when [tex]\( x = 0 \)[/tex].
- For negative [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] yields values less than 1, while [tex]\( g(x) \)[/tex] yields values greater than 1.
- Essentially, [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inversely related and symmetrical about the [tex]\( y \)[/tex]-axis.
Therefore, the relationship between the functions can be concluded as:
The functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are reflections over the [tex]\( y \)[/tex] axis.
