Answered

Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

The [tex]\( x \)[/tex]-values in the table for [tex]\( f(x) \)[/tex] were multiplied by -1 to create the table for [tex]\( g(x) \)[/tex]. What is the relationship between the graphs of the two functions?

[tex]\[
\begin{tabular}{|c|c|}
\hline
\multicolumn{1}{|c|}{$f(x)$} \\
\hline
-2 & -31 \\
\hline
-1 & 0 \\
\hline
1 & 2 \\
\hline
2 & 33 \\
\hline
\end{tabular}
\begin{tabular}{|c|c|}
\hline
\multicolumn{1}{c}{$g(x)$} \\
\hline
2 & $y$ \\
\hline
1 & -31 \\
\hline
1 & 0 \\
\hline
-1 & 2 \\
\hline
-2 & 33 \\
\hline
\end{tabular}
\][/tex]

A. They are reflections of each other across the [tex]\( x \)[/tex]-axis.
B. They are reflections of each other over the line [tex]\( x=y \)[/tex].
C. They are reflections of each other across the [tex]\( y \)[/tex]-axis.
D. The graphs are not related.

Sagot :

GinnyAnswer
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.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.