Answer :
To determine how \( f(x) \) was transformed to create \( g(x) \), we need to analyze the given table of values. Let's look at how the corresponding \( f(x) \) and \( g(x) \) values change:
For \( x = -2 \):
\( f(-2) = \frac{1}{9} \)
\( g(-2) = \frac{1}{81} = \left( \frac{1}{9} \right)^2 \)
For \( x = -1 \):
\( f(-1) = \frac{1}{3} \)
\( g(-1) = \frac{1}{27} = \left( \frac{1}{3} \right)^2 \)
For \( x = 2 \):
\( f(2) = 9 \)
\( g(2) = 1 = \left( 9 \right)^{1/2} \)
For \( x = 3 \):
\( f(3) = 27 \)
\( g(3) = 3 = \left( 27 \right)^{1/3} \)
For \( x = 4 \):
\( f(4) = 81 \)
\( g(4) = 9 = \left( 81 \right)^{1/2} \)
From these observations, we see that the values of \( g(x) \) are obtained by taking either the square or the cube root of \( f(x) \):
- For \( x = -2 \) and \( x = -1 \), \( g(x) = (f(x))^2 \)
- For \( x = 2 \) and \( x = 4 \), \( g(x) = (f(x))^{1/2} \)
Combining these observations for a consistent transformation across all values:
- The transformation is most consistently represented by \( g(x) = (f(x))^{1/2} \).
Given that all values of \( g(x) \) correspond to the square root or power of \( f(x) \), we can conclude that the transformation applied here is a vertical stretch by a factor represented by the exponent applied to \( f(x) \).
The answer is:
D. Horizontal or vertical stretch
For \( x = -2 \):
\( f(-2) = \frac{1}{9} \)
\( g(-2) = \frac{1}{81} = \left( \frac{1}{9} \right)^2 \)
For \( x = -1 \):
\( f(-1) = \frac{1}{3} \)
\( g(-1) = \frac{1}{27} = \left( \frac{1}{3} \right)^2 \)
For \( x = 2 \):
\( f(2) = 9 \)
\( g(2) = 1 = \left( 9 \right)^{1/2} \)
For \( x = 3 \):
\( f(3) = 27 \)
\( g(3) = 3 = \left( 27 \right)^{1/3} \)
For \( x = 4 \):
\( f(4) = 81 \)
\( g(4) = 9 = \left( 81 \right)^{1/2} \)
From these observations, we see that the values of \( g(x) \) are obtained by taking either the square or the cube root of \( f(x) \):
- For \( x = -2 \) and \( x = -1 \), \( g(x) = (f(x))^2 \)
- For \( x = 2 \) and \( x = 4 \), \( g(x) = (f(x))^{1/2} \)
Combining these observations for a consistent transformation across all values:
- The transformation is most consistently represented by \( g(x) = (f(x))^{1/2} \).
Given that all values of \( g(x) \) correspond to the square root or power of \( f(x) \), we can conclude that the transformation applied here is a vertical stretch by a factor represented by the exponent applied to \( f(x) \).
The answer is:
D. Horizontal or vertical stretch
