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Answer :
To figure out which table represents the inverse of the given function, we need to understand what an inverse function is. For a function, the inverse involves swapping the input and output values. Basically, if you have a function where [tex]\( x \)[/tex] maps to [tex]\( y \)[/tex], in the inverse function, [tex]\( y \)[/tex] will map back to [tex]\( x \)[/tex].
Let's apply this idea to the table provided:
Original Function Table:
- [tex]\( x: 7, 10, 13, 16 \)[/tex]
- [tex]\( y: 21, 30, 39, 48 \)[/tex]
For the inverse, we switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. So the [tex]\( y \)[/tex]-values in the original function become the [tex]\( x \)[/tex]-values in the inverse, and the [tex]\( x \)[/tex]-values of the original become the [tex]\( y \)[/tex]-values in the inverse.
Inverse Function Table:
- [tex]\( x: 21, 30, 39, 48 \)[/tex]
- [tex]\( y: 7, 10, 13, 16 \)[/tex]
Now, let's compare this to the answer choices given:
- Option A:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline$x$ & 21 & 30 & 39 & 48 \\
\hline$y$ & 7 & 10 & 13 & 16 \\
\hline
\end{tabular}
\][/tex]
This option matches the inverse table we deduced from the original function.
Options B, C, and D do not satisfy the condition because they do not swap the input and output values properly or introduce incorrect signs.
Therefore, the correct table representing the inverse of the original function is Option A.
Let's apply this idea to the table provided:
Original Function Table:
- [tex]\( x: 7, 10, 13, 16 \)[/tex]
- [tex]\( y: 21, 30, 39, 48 \)[/tex]
For the inverse, we switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. So the [tex]\( y \)[/tex]-values in the original function become the [tex]\( x \)[/tex]-values in the inverse, and the [tex]\( x \)[/tex]-values of the original become the [tex]\( y \)[/tex]-values in the inverse.
Inverse Function Table:
- [tex]\( x: 21, 30, 39, 48 \)[/tex]
- [tex]\( y: 7, 10, 13, 16 \)[/tex]
Now, let's compare this to the answer choices given:
- Option A:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline$x$ & 21 & 30 & 39 & 48 \\
\hline$y$ & 7 & 10 & 13 & 16 \\
\hline
\end{tabular}
\][/tex]
This option matches the inverse table we deduced from the original function.
Options B, C, and D do not satisfy the condition because they do not swap the input and output values properly or introduce incorrect signs.
Therefore, the correct table representing the inverse of the original function is Option A.
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