Answer :
To determine the relationship between the graphs of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] based on the given tables, we need to analyze the transformation that has been applied to the [tex]\( x \)[/tex]-values of [tex]\( f(x) \)[/tex] to produce [tex]\( g(x) \)[/tex]:
The table for [tex]\( f(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline \text{x} & \text{f(x)} \\ \hline -2 & -31 \\ \hline -1 & 0 \\ \hline 1 & 2 \\ \hline 2 & 33 \\ \hline \end{array} \][/tex]
The table for [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline \text{x} & \text{y (g(x))} \\ \hline 2 & -31 \\ \hline 1 & 0 \\ \hline -1 & 2 \\ \hline -2 & 33 \\ \hline \end{array} \][/tex]
First, we observe how the [tex]\( x \)[/tex]-values in the [tex]\( g(x) \)[/tex] table correspond to the [tex]\( x \)[/tex]-values in the [tex]\( f(x) \)[/tex] table:
- When [tex]\( f(x) \)[/tex] has [tex]\( x = -2 \)[/tex], [tex]\( g(x) \)[/tex] has [tex]\( x = 2 \)[/tex].
- When [tex]\( f(x) \)[/tex] has [tex]\( x = -1 \)[/tex], [tex]\( g(x) \)[/tex] has [tex]\( x = 1 \)[/tex].
- When [tex]\( f(x) \)[/tex] has [tex]\( x = 1 \)[/tex], [tex]\( g(x) \)[/tex] has [tex]\( x = -1 \)[/tex].
- When [tex]\( f(x) \)[/tex] has [tex]\( x = 2 \)[/tex], [tex]\( g(x) \)[/tex] has [tex]\( x = -2 \)[/tex].
We can see that each [tex]\( x \)[/tex]-value in [tex]\( f(x) \)[/tex] has been multiplied by [tex]\(-1\)[/tex] to become the corresponding [tex]\( x \)[/tex]-value in [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = f(-x) \][/tex]
This transformation [tex]\( g(x) = f(-x) \)[/tex] is a reflection of [tex]\( f(x) \)[/tex] across the [tex]\( y \)[/tex]-axis.
Let's consider the options provided:
A. Reflections across the [tex]\( x \)[/tex]-axis would mean the [tex]\( y \)[/tex]-values get negated. This means [tex]\( g(x) = -f(x) \)[/tex], which is not the case here.
B. Reflections over the line [tex]\( x=y \)[/tex] imply swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the points. This transformation would not apply in this scenario.
C. Reflections across the [tex]\( y \)[/tex]-axis mean the [tex]\( x \)[/tex]-values get negated. This means [tex]\( g(x) = f(-x) \)[/tex], which is what we have demonstrated.
D. No relationship would mean they don't directly reflect or transform, which is not the case here.
Thus, the correct relationship is:
C. They are reflections of each other across the [tex]\( y \)[/tex]-axis.
The table for [tex]\( f(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline \text{x} & \text{f(x)} \\ \hline -2 & -31 \\ \hline -1 & 0 \\ \hline 1 & 2 \\ \hline 2 & 33 \\ \hline \end{array} \][/tex]
The table for [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline \text{x} & \text{y (g(x))} \\ \hline 2 & -31 \\ \hline 1 & 0 \\ \hline -1 & 2 \\ \hline -2 & 33 \\ \hline \end{array} \][/tex]
First, we observe how the [tex]\( x \)[/tex]-values in the [tex]\( g(x) \)[/tex] table correspond to the [tex]\( x \)[/tex]-values in the [tex]\( f(x) \)[/tex] table:
- When [tex]\( f(x) \)[/tex] has [tex]\( x = -2 \)[/tex], [tex]\( g(x) \)[/tex] has [tex]\( x = 2 \)[/tex].
- When [tex]\( f(x) \)[/tex] has [tex]\( x = -1 \)[/tex], [tex]\( g(x) \)[/tex] has [tex]\( x = 1 \)[/tex].
- When [tex]\( f(x) \)[/tex] has [tex]\( x = 1 \)[/tex], [tex]\( g(x) \)[/tex] has [tex]\( x = -1 \)[/tex].
- When [tex]\( f(x) \)[/tex] has [tex]\( x = 2 \)[/tex], [tex]\( g(x) \)[/tex] has [tex]\( x = -2 \)[/tex].
We can see that each [tex]\( x \)[/tex]-value in [tex]\( f(x) \)[/tex] has been multiplied by [tex]\(-1\)[/tex] to become the corresponding [tex]\( x \)[/tex]-value in [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = f(-x) \][/tex]
This transformation [tex]\( g(x) = f(-x) \)[/tex] is a reflection of [tex]\( f(x) \)[/tex] across the [tex]\( y \)[/tex]-axis.
Let's consider the options provided:
A. Reflections across the [tex]\( x \)[/tex]-axis would mean the [tex]\( y \)[/tex]-values get negated. This means [tex]\( g(x) = -f(x) \)[/tex], which is not the case here.
B. Reflections over the line [tex]\( x=y \)[/tex] imply swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the points. This transformation would not apply in this scenario.
C. Reflections across the [tex]\( y \)[/tex]-axis mean the [tex]\( x \)[/tex]-values get negated. This means [tex]\( g(x) = f(-x) \)[/tex], which is what we have demonstrated.
D. No relationship would mean they don't directly reflect or transform, which is not the case here.
Thus, the correct relationship is:
C. They are reflections of each other across the [tex]\( y \)[/tex]-axis.
