To determine which transformation of [tex]\( f(x) \)[/tex] will produce [tex]\( g(x) \)[/tex], let's analyze the given table of values.
The given values are:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
-8 & 16 & 4 \\
\hline
-4 & 4 & 1 \\
\hline
0 & 0 & 0 \\
\hline
4 & 4 & 1 \\
\hline
8 & 16 & 4 \\
\hline
\end{array}
\][/tex]
Now, let's consider each proposed transformation and evaluate whether it matches the given table of values:
1. [tex]\( g(x) = \frac{1}{4} f(x) \)[/tex]
- For [tex]\( x = -8 \)[/tex]: [tex]\(\frac{1}{4} f(-8) = \frac{1}{4} \times 16 = 4\)[/tex]
- For [tex]\( x = -4 \)[/tex]: [tex]\(\frac{1}{4} f(-4) = \frac{1}{4} \times 4 = 1\)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\(\frac{1}{4} f(0) = \frac{1}{4} \times 0 = 0\)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\(\frac{1}{4} f(4) = \frac{1}{4} \times 4 = 1\)[/tex]
- For [tex]\( x = 8 \)[/tex]: [tex]\(\frac{1}{4} f(8) = \frac{1}{4} \times 16 = 4\)[/tex]
We see that these values exactly match the [tex]\( g(x) \)[/tex] values in the table. Therefore, [tex]\( g(x) = \frac{1}{4} f(x) \)[/tex] is a valid transformation.
2. [tex]\( g(x) = f\left(\frac{1}{4} x\right) \)[/tex]
- For [tex]\( x = -8 \)[/tex]: [tex]\( f\left(\frac{-8}{4}\right) = f(-2) \)[/tex], but [tex]\( f(-2) \)[/tex] is not provided in the table, so we cannot verify directly.
- Evaluators for other [tex]\( x \)[/tex] values would similarly work via [tex]\( f \)[/tex] at non-provided [tex]\( x \)[/tex] values; hence we cannot use this transformation from the given data.
Given the lack of transformation verification from table data, this cannot be concluded as correct.
3. [tex]\( g(x) = 4 f(x) \)[/tex]
- For [tex]\( x = -8 \)[/tex]: [tex]\( 4 f(-8) = 4 \times 16 = 64 \)[/tex]
- For [tex]\( x = -4 \)[/tex]: [tex]\( 4 f(-4) = 4 \times 4 = 16 \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( 4 f(0) = 4 \times 0 = 0 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( 4 f(4) = 4 \times 4 = 16 \)[/tex]
- For [tex]\( x = 8 \)[/tex]: [tex]\( 4 f(8) = 4 \times 16 = 64 \)[/tex]
These values do not match the [tex]\( g(x) \)[/tex] values in the table. Therefore, [tex]\( g(x) = 4 f(x) \)[/tex] is not the correct transformation.
4. [tex]\( g(x) = f(4x) \)[/tex]
- For [tex]\( x = -8 \)[/tex]: [tex]\( f(4 \times -8) = f(-32) \)[/tex], which is not provided.
- Similarly, other [tex]\( x \)[/tex] transformations would lead to [tex]\( f \)[/tex] at non-provided values.
This cannot be validated via provided [tex]\( x \)[/tex] table data.
Given the valid matching computed in the first transformation:
The correct transformation is [tex]\( g(x) = \frac{1}{4} f(x) \)[/tex].
