Question

the function $f(x)$ is defined as $f(x) = \frac{1}{3}(6)^x$. which table of values could be used to graph $g(x)$, a reflection of $f(x)$ across the $x$-axis? tables of values are shown with x, f(x), g(x) columns for different x values (-2, -1, 0, 1, 2) and corresponding f(x) and g(x) values in each table

Explanation:

Step1: Recall Reflection Rule

Reflecting a function \( f(x) \) across the \( x \)-axis gives \( g(x)= -f(x) \). So for each \( x \), \( g(x) \) should be the negative of \( f(x) \).

Step2: Analyze \( f(x) \) at \( x = 0 \)

Calculate \( f(0)=\frac{1}{3}(6)^0=\frac{1}{3}(1)=\frac{1}{3} \). So \( g(0)= -f(0)=-\frac{1}{3} \).

Step3: Check Tables for \( x = 0 \)

• First table: \( f(0)=0 \) (incorrect, since \( f(0)=\frac{1}{3} \)), so eliminate.
• Second table: \( f(0)=0 \) (incorrect), eliminate.
• Third table: \( f(0)=\frac{1}{3} \), but \( g(0)=\frac{1}{3} \) (not negative), eliminate.
• Fourth table: \( f(0)=\frac{1}{3} \), \( g(0)=-\frac{1}{3} \) (matches \( g(x)=-f(x) \)). Check other values:
• \( x=-2 \): \( f(-2)=\frac{1}{108} \), \( g(-2)=-\frac{1}{108} \) (matches).
• \( x=-1 \): \( f(-1)=\frac{1}{18} \), \( g(-1)=-\frac{1}{18} \) (matches).
• \( x=1 \): \( f(1)=2 \), \( g(1)=-2 \) (matches).
• \( x=2 \): \( f(2)=12 \), \( g(2)=-12 \) (matches).

Answer:

The table with blue header (the fourth table, bottom - left) where for \( x=-2, -1, 0, 1, 2 \), \( g(x) \) is the negative of \( f(x) \).