To determine how the function [tex]\( f(x) \)[/tex] was transformed to become [tex]\( g(x) \)[/tex], we need to analyze the given pairs of [tex]\((x, y)\)[/tex] points for both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. Let's compare the [tex]\( y \)[/tex]-values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] for corresponding [tex]\( x \)[/tex]-values.
Given points:
- For [tex]\( x = -2 \)[/tex]:
[tex]\( f(-2) = \frac{1}{9} \)[/tex]
[tex]\( g(-2) = -\frac{17}{9} \)[/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\( f(-1) = \frac{1}{3} \)[/tex]
[tex]\( g(-1) = -\frac{5}{3} \)[/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\( f(2) = 9 \)[/tex]
[tex]\( g(2) = 7 \)[/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\( f(3) = 27 \)[/tex]
[tex]\( g(3) = 25 \)[/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\( f(4) = 81 \)[/tex]
[tex]\( g(4) = 79 \)[/tex]
From this, we can determine how [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] relate to each other for each point:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = f(-2) - 2 \quad \Rightarrow \quad -\frac{17}{9} = \frac{1}{9} - 2 \][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = f(-1) - 2 \quad \Rightarrow \quad -\frac{5}{3} = \frac{1}{3} - 2 \][/tex]
3. For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = f(2) - 2 \quad \Rightarrow \quad 7 = 9 - 2 \][/tex]
4. For [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = f(3) - 2 \quad \Rightarrow \quad 25 = 27 - 2 \][/tex]
5. For [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = f(4) - 2 \quad \Rightarrow \quad 79 = 81 - 2 \][/tex]
It is clear from these calculations that for any [tex]\( x \)[/tex]:
[tex]\[ g(x) = f(x) - 2 \][/tex]
This describes a vertical shift of the parent function [tex]\( f(x) \)[/tex] downward by 2 units.
Therefore, the correct transformation is a vertical shift.
Thus, the correct answer is:
[tex]\[
\boxed{4 \text{ (Horizontal or vertical shift)}}
\][/tex]
