To determine which table represents the inverse of the function given above, we need to recall that the inverse of a function swaps the roles of the input (x-values) and output (y-values). Essentially, for the inverse, what was originally the y-coordinate becomes the x-coordinate and vice versa.
Given the original function:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 7 & 10 & 13 & 16 \\
\hline
y & 21 & 30 & 39 & 48 \\
\hline
\end{array}
\][/tex]
To find the inverse, swap the x and y columns:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 21 & 30 & 39 & 48 \\
\hline
y & 7 & 10 & 13 & 16 \\
\hline
\end{array}
\][/tex]
Now, let's match this with the provided options:
A.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -7 & -10 & -13 & -16 \\
\hline
y & 21 & 30 & 39 & 48 \\
\hline
\end{array}
\][/tex]
Here the [tex]\(x\)[/tex]-values are simply the negative of the original [tex]\(x\)[/tex]-values, and the [tex]\(y\)[/tex]-values remain the same. This does not represent the inverse.
B.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -21 & -30 & -39 & -48 \\
\hline
y & 7 & 10 & 13 & 16 \\
\hline
\end{array}
\][/tex]
Here, the [tex]\(x\)[/tex]-values are the negative of the original [tex]\(y\)[/tex]-values, and the [tex]\(y\)[/tex]-values are the original [tex]\(x\)[/tex]-values. This does not represent the inverse.
C.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 7 & 10 & 13 & 16 \\
\hline
y & -21 & -30 & -39 & -48 \\
\hline
\end{array}
\][/tex]
Here, the [tex]\(x\)[/tex]-values remain the same, and the [tex]\(y\)[/tex]-values are the negative of the original [tex]\(y\)[/tex]-values. This also does not represent the inverse.
D.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 21 & 30 & 39 & 48 \\
\hline
y & 7 & 10 & 13 & 16 \\
\hline
\end{array}
\][/tex]
Here, we have exactly swapped the original [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values. This table correctly represents the inverse of the given function.
Therefore, the correct answer is:
D.
