To determine which of the given tables represents the values of [tex]\( g(x) \)[/tex], the reflection of [tex]\( f(x) \)[/tex] across the [tex]\( x \)[/tex]-axis, we need to understand that reflecting [tex]\( f(x) \)[/tex] across the [tex]\( x \)[/tex]-axis results in [tex]\( g(x) = -f(x) \)[/tex].
Given the function [tex]\( f(x) = \frac{1}{3} \left( 6 \right)^x \)[/tex], we will perform the reflection to find [tex]\( g(x) \)[/tex].
Step-by-step calculation:
1. Define [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = \frac{1}{3} \left( 6 \right)^x \][/tex]
2. Calculate values of [tex]\( f(x) \)[/tex] for given [tex]\( x \)[/tex] values:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = \frac{1}{3} \left( 6 \right)^{-2} = \frac{1}{3} \cdot \frac{1}{36} = \frac{1}{108} \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \frac{1}{3} \left( 6 \right)^{-1} = \frac{1}{3} \cdot \frac{1}{6} = \frac{1}{18} \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{1}{3} \left( 6 \right)^0 = \frac{1}{3} \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \frac{1}{3} \left( 6 \right)^1 = \frac{6}{3} = 2 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} \left( 6 \right)^2 = \frac{36}{3} = 12 \][/tex]
The corresponding values of [tex]\( f(x) \)[/tex] are:
[tex]\[ \begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & \frac{1}{108} \\
-1 & \frac{1}{18} \\
0 & \frac{1}{3} \\
1 & 2 \\
2 & 12 \\
\hline
\end{array} \][/tex]
3. Calculate [tex]\( g(x) = -f(x) \)[/tex]:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = -f(-2) = -\frac{1}{108} \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = -f(-1) = -\frac{1}{18} \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -f(0) = -\frac{1}{3} \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -f(1) = -2 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = -f(2) = -12 \][/tex]
4. Check the two given tables to see which matches [tex]\( g(x) \)[/tex]:
Comparing the values of [tex]\( g(x) \)[/tex] in the two tables:
- Table 1 does not match as the values in [tex]\( g(x) \)[/tex] column are inverted and not the ones calculated.
- Table 2 also does not match because it has incorrect values including a zero.
Given these observations, the corresponding [tex]\( g(x) \)[/tex] values do not match with either table presented.
