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Consider the tables created using an initial investment of \[tex]$1,000 and quarterly compounding of interest.

Table A represents the function that models the total amount of one investment, \(a(x)\), based on the annual interest rate, \(x\), as a percent.

Table B represents the function that models the interest rate, \(r(d)\), as a percent, based on the total amount at the end of the investment, \(d\).

\ \textless \ strong\ \textgreater \ Table A\ \textless \ /strong\ \textgreater \
\[
\begin{tabular}{|c|c|c|c|}
\hline
$[/tex]x[tex]$ & 0.75 & 1 & 1.25 \\
\hline
$[/tex]a(x)[tex]$ & \$[/tex]1,038.18 & \[tex]$1,051.21 & \$[/tex]1,064.39 \\
\hline
\end{tabular}
\]

Table B
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$d$ & \$1,057.81 & \$1,077.78 & \$1,098.12 \\
\hline
$r(d)$ & 0.75 & 1 & 1.25 \\
\hline
\end{tabular}
\][/tex]

Use the values in the table to verify the relationship between the functions representing the investments.

Which conclusion can be made?
A. The functions are inverses because the domain of Table A is the same as the range of Table B.
B. The functions are inverses because the range of Table A is different from the domain of Table B.
C. The functions are not inverses because for each ordered pair [tex]\((x, y)\)[/tex] for one function, there is no corresponding ordered pair [tex]\((x, y)\)[/tex] for the other function.
D. The functions are not inverses because for each ordered pair [tex]\((x, y)\)[/tex] for one function, there is no corresponding ordered pair [tex]\((y, x)\)[/tex] for the other function.



Answer :

GinnyAnswer
To verify whether the functions represented by Table A and Table B are inverses of each other, we need to check the correspondence between the pairs in the tables.

Table A
[tex]\[ \begin{array}{|c|c|c|c|} \hline x & 0.75 & 1 & 1.25 \\ \hline a(x) & 1038.18 & 1051.21 & 1064.39 \\ \hline \end{array} \][/tex]

Table B
[tex]\[ \begin{array}{|c|c|c|c|} \hline d & 1057.81 & 1077.78 & 1098.12 \\ \hline r(d) & 0.75 & 1 & 1.25 \\ \hline \end{array} \][/tex]

To determine if the functions are inverses, we need to verify if for each [tex]\((x, a(x))\)[/tex] pair in Table A, there is a corresponding [tex]\((a(x), x)\)[/tex] pair in Table B. This means:

- For [tex]\(x = 0.75\)[/tex] in Table A:
- [tex]\(a(x) = 1038.18\)[/tex]
- Check if [tex]\(d = 1038.18\)[/tex] in Table B and [tex]\(r(d) = 0.75\)[/tex].
- In Table B, [tex]\(d = 1038.18\)[/tex] is not present.

- For [tex]\(x = 1\)[/tex] in Table A:
- [tex]\(a(x) = 1051.21\)[/tex]
- Check if [tex]\(d = 1051.21\)[/tex] in Table B and [tex]\(r(d) = 1\)[/tex].
- In Table B, [tex]\(d = 1051.21\)[/tex] is not present.

- For [tex]\(x = 1.25\)[/tex] in Table A:
- [tex]\(a(x) = 1064.39\)[/tex]
- Check if [tex]\(d = 1064.39\)[/tex] in Table B and [tex]\(r(d) = 1.25\)[/tex].
- In Table B, [tex]\(d = 1064.39\)[/tex] is not present.

Since there are no corresponding [tex]\( (d, r(d)) \)[/tex] pairs in Table B for the [tex]\((x, a(x))\)[/tex] pairs in Table A, we conclude that the functions are not inverses.

Thus, the correct conclusion is:
"The functions are not inverses because for each ordered pair [tex]\((x, y)\)[/tex] for one function, there is no corresponding ordered pair [tex]\((y, x)\)[/tex] for the other function."

Hence, the answer would be:
"The functions are not inverses because for each ordered pair [tex]\((x, y)\)[/tex] for one function, there is no corresponding ordered pair [tex]\((y, x)\)[/tex] for the other function."

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