Answer :
To determine how the function [tex]\( f(x) \)[/tex] was transformed to create [tex]\( g(x) \)[/tex], let’s analyze the given data in detail.
We are provided with the following table:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \multicolumn{2}{|c|}{f(x)} & \multicolumn{2}{|c|}{g(x)} \\ \hline x & y & x & y \\ \hline -2 & \frac{1}{9} & -2 & \frac{1}{81} \\ \hline -1 & \frac{1}{3} & -1 & \frac{1}{27} \\ \hline 2 & 9 & 2 & 1 \\ \hline 3 & 27 & 3 & 3 \\ \hline 4 & 81 & 4 & 9 \\ \hline \end{array} \][/tex]
We need to decide on the type of transformation that has been applied to [tex]\( f(x) \)[/tex] to produce [tex]\( g(x) \)[/tex].
### Step-by-Step Analysis:
1. Identify the relationship between the corresponding values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- When [tex]\(x = -2\)[/tex]:
[tex]\( f(x) = \frac{1}{9} \)[/tex]
[tex]\( g(x) = \frac{1}{81} \)[/tex]
- When [tex]\(x = -1\)[/tex]:
[tex]\( f(x) = \frac{1}{3} \)[/tex]
[tex]\( g(x) = \frac{1}{27} \)[/tex]
- When [tex]\(x = 2\)[/tex]:
[tex]\( f(x) = 9 \)[/tex]
[tex]\( g(x) = 1 \)[/tex]
- When [tex]\(x = 3\)[/tex]:
[tex]\( f(x) = 27 \)[/tex]
[tex]\( g(x) = 3 \)[/tex]
- When [tex]\(x = 4\)[/tex]:
[tex]\( f(x) = 81 \)[/tex]
[tex]\( g(x) = 9 \)[/tex]
2. Calculate the ratios [tex]\( \frac{f(x)}{g(x)} \)[/tex] to determine the consistency of transformation:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ \frac{f(-2)}{g(-2)} = \frac{\frac{1}{9}}{\frac{1}{81}} = \frac{1}{9} \times \frac{81}{1} = 9 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ \frac{f(-1)}{g(-1)} = \frac{\frac{1}{3}}{\frac{1}{27}} = \frac{1}{3} \times \frac{27}{1} = 9 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ \frac{f(2)}{g(2)} = \frac{9}{1} = 9 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ \frac{f(3)}{g(3)} = \frac{27}{3} = 9 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ \frac{f(4)}{g(4)} = \frac{81}{9} = 9 \][/tex]
The ratio [tex]\(\frac{f(x)}{g(x)}\)[/tex] is consistently [tex]\( 9 \)[/tex].
3. Interpretation of the consistent ratio:
Since the ratio of [tex]\(\frac{f(x)}{g(x)}\)[/tex] is [tex]\(9\)[/tex] for all values of [tex]\(x\)[/tex], this indicates that [tex]\(g(x)\)[/tex] is a result of a vertical compression of [tex]\(f(x)\)[/tex] by a factor of [tex]\( \frac{1}{9} \)[/tex].
Vertical compression means that each [tex]\( y \)[/tex]-value of [tex]\( f(x) \)[/tex] is multiplied by [tex]\( \frac{1}{9} \)[/tex] to get the corresponding [tex]\( y \)[/tex]-value in [tex]\( g(x) \)[/tex].
Therefore, the transformation type is:
B. Horizontal or vertical stretch
Since the actual transformation observed is a vertical compression (which is a type of vertical stretch but in the opposite manner), this is the most accurate description among the given options.
We are provided with the following table:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \multicolumn{2}{|c|}{f(x)} & \multicolumn{2}{|c|}{g(x)} \\ \hline x & y & x & y \\ \hline -2 & \frac{1}{9} & -2 & \frac{1}{81} \\ \hline -1 & \frac{1}{3} & -1 & \frac{1}{27} \\ \hline 2 & 9 & 2 & 1 \\ \hline 3 & 27 & 3 & 3 \\ \hline 4 & 81 & 4 & 9 \\ \hline \end{array} \][/tex]
We need to decide on the type of transformation that has been applied to [tex]\( f(x) \)[/tex] to produce [tex]\( g(x) \)[/tex].
### Step-by-Step Analysis:
1. Identify the relationship between the corresponding values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- When [tex]\(x = -2\)[/tex]:
[tex]\( f(x) = \frac{1}{9} \)[/tex]
[tex]\( g(x) = \frac{1}{81} \)[/tex]
- When [tex]\(x = -1\)[/tex]:
[tex]\( f(x) = \frac{1}{3} \)[/tex]
[tex]\( g(x) = \frac{1}{27} \)[/tex]
- When [tex]\(x = 2\)[/tex]:
[tex]\( f(x) = 9 \)[/tex]
[tex]\( g(x) = 1 \)[/tex]
- When [tex]\(x = 3\)[/tex]:
[tex]\( f(x) = 27 \)[/tex]
[tex]\( g(x) = 3 \)[/tex]
- When [tex]\(x = 4\)[/tex]:
[tex]\( f(x) = 81 \)[/tex]
[tex]\( g(x) = 9 \)[/tex]
2. Calculate the ratios [tex]\( \frac{f(x)}{g(x)} \)[/tex] to determine the consistency of transformation:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ \frac{f(-2)}{g(-2)} = \frac{\frac{1}{9}}{\frac{1}{81}} = \frac{1}{9} \times \frac{81}{1} = 9 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ \frac{f(-1)}{g(-1)} = \frac{\frac{1}{3}}{\frac{1}{27}} = \frac{1}{3} \times \frac{27}{1} = 9 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ \frac{f(2)}{g(2)} = \frac{9}{1} = 9 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ \frac{f(3)}{g(3)} = \frac{27}{3} = 9 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ \frac{f(4)}{g(4)} = \frac{81}{9} = 9 \][/tex]
The ratio [tex]\(\frac{f(x)}{g(x)}\)[/tex] is consistently [tex]\( 9 \)[/tex].
3. Interpretation of the consistent ratio:
Since the ratio of [tex]\(\frac{f(x)}{g(x)}\)[/tex] is [tex]\(9\)[/tex] for all values of [tex]\(x\)[/tex], this indicates that [tex]\(g(x)\)[/tex] is a result of a vertical compression of [tex]\(f(x)\)[/tex] by a factor of [tex]\( \frac{1}{9} \)[/tex].
Vertical compression means that each [tex]\( y \)[/tex]-value of [tex]\( f(x) \)[/tex] is multiplied by [tex]\( \frac{1}{9} \)[/tex] to get the corresponding [tex]\( y \)[/tex]-value in [tex]\( g(x) \)[/tex].
Therefore, the transformation type is:
B. Horizontal or vertical stretch
Since the actual transformation observed is a vertical compression (which is a type of vertical stretch but in the opposite manner), this is the most accurate description among the given options.
