To determine the value of [tex]\( k \)[/tex] for which [tex]\( g(x) = f(kx) \)[/tex], given the table of values for [tex]\( g(x) \)[/tex], we can follow these steps:
1. Observe the Table Values for [tex]\( g(x) \)[/tex]:
The table gives us the following values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-4 & 4 \\
\hline
-2 & 1 \\
\hline
0 & 0 \\
\hline
2 & 1 \\
\hline
4 & 4 \\
\hline
\end{array}
\][/tex]
2. Recognize the Symmetry in [tex]\( g(x) \)[/tex]:
The values of [tex]\( g(x) \)[/tex] (specifically, [tex]\( g(-x) = g(x) \)[/tex]) indicate a symmetry. It suggests that the corresponding function [tex]\( f \)[/tex] may follow a similar pattern.
3. Identify Key Points:
Notice that [tex]\( g(4) = g(-4) = 4 \)[/tex] and [tex]\( g(2) = g(-2) = 1 \)[/tex]. These correspondences will help us determine [tex]\( k \)[/tex].
4. Matching [tex]\( g(x) = f(kx) \)[/tex] with [tex]\( g(x) \)[/tex] Points:
- For [tex]\( x = 2 \)[/tex]: [tex]\( g(2) = 1 \)[/tex], which means [tex]\( f(k \cdot 2) = 1 \)[/tex].
- For [tex]\( x = 4 \)[/tex]: [tex]\( g(4) = 4 \)[/tex], which means [tex]\( f(k \cdot 4) = 4 \)[/tex].
5. Assuming [tex]\( k \)[/tex] Reduces [tex]\( x \)[/tex] Proportionally:
We need to find a [tex]\( k \)[/tex] such that:
- When [tex]\( x = 2 \)[/tex], [tex]\( f(2k) = 1 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( f(4k) = 4 \)[/tex]
6. Determine [tex]\( k \)[/tex] by Matching Transformed Points:
Given that [tex]\( f \)[/tex] achieves its corresponding outcomes at these points, we can infer a transformation factor. We specifically see that for [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex]:
- [tex]\( 2k \)[/tex] should correspond to the [tex]\( x \)[/tex] value of 1 for another function [tex]\( f \)[/tex].
- [tex]\( 4k \)[/tex] should correspond to the [tex]\( x \)[/tex] value of 2 for another function [tex]\( f \)[/tex].
If we examine these relationships, [tex]\( k \)[/tex] effectively compresses or stretches [tex]\( x \)[/tex] into desired points.
- If [tex]\( 2k = 1 \)[/tex], then [tex]\( k = \frac{1}{2} \)[/tex].
Therefore, aligning the points [tex]\( f(kx) \)[/tex] with the observed [tex]\( g(x) \)[/tex] values dictates:
[tex]\[ k = \frac{1}{2} \][/tex]
The value of [tex]\( k \)[/tex] is [tex]\( 0.5 \)[/tex].
