To determine the relationship between the functions [tex]\( f(x) = 2^x \)[/tex] and [tex]\( g(x) = \left( \frac{1}{2} \right)^x \)[/tex] based on the given table, let's examine the data points provided:
[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]
1. Reflection Over the x-axis:
For two functions to be reflections over the [tex]\( x \)[/tex]-axis, [tex]\( f(x) \)[/tex] would need to be the negative of [tex]\( g(x) \)[/tex] for every [tex]\( x \)[/tex]. For instance, [tex]\( f(x) = -g(x) \)[/tex]. But here, we see that [tex]\( 2^x \)[/tex] and [tex]\( \left( \frac{1}{2} \right)^x \)[/tex] do not satisfy this criterion. For example, [tex]\( f(2) = 4 \)[/tex] and [tex]\( g(2) = \frac{1}{4} \)[/tex], which are not negatives of each other.
2. Reflection Over the y-axis:
For two functions to be reflections over the [tex]\( y \)[/tex]-axis, [tex]\( f(x) \)[/tex] would need to be equal to [tex]\( g(-x) \)[/tex] for every [tex]\( x \)[/tex]. Check the table: [tex]\( f(2) = 4 \)[/tex] and [tex]\( g(-2) = 4 \)[/tex]; [tex]\( f(1) = 2 \)[/tex] and [tex]\( g(-1) = 2 \)[/tex]; [tex]\( f(0) = 1 \)[/tex] and [tex]\( g(0) = 1 \)[/tex]; [tex]\( f(-1) = \frac{1}{2} \)[/tex] and [tex]\( g(1) = \frac{1}{2} \)[/tex]; [tex]\( f(-2) = \frac{1}{4} \)[/tex] and [tex]\( g(2) = \frac{1}{4} \)[/tex]. This shows that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are indeed reflections over the y-axis.
3. Increasing vs. Decreasing:
A function [tex]\( f(x) \)[/tex] is increasing if as [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] also increases. Similarly, a function [tex]\( g(x) \)[/tex] is decreasing if as [tex]\( x \)[/tex] increases, [tex]\( g(x) \)[/tex] decreases. Check the values: as [tex]\( x \)[/tex] increases, [tex]\( f(x) = 2^x \)[/tex] increases (4, 2, 1, 1/2, 1/4) and [tex]\( g(x) = \left( \frac{1}{2} \right)^x \)[/tex] decreases (1/4, 1/2, 1, 2, 4). Therefore, [tex]\( f(x) \)[/tex] is increasing and [tex]\( g(x) \)[/tex] is decreasing, not the other way around.
4. Initial Values:
The initial value usually refers to the function value at [tex]\( x = 0 \)[/tex]. Both [tex]\( f(x) = 2^0 = 1 \)[/tex] and [tex]\( g(x) = \left( \frac{1}{2} \right)^0 = 1 \)[/tex] have the same initial value.
Given these observations, the correct conclusion is:
The functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are reflections over the y-axis.
