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

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

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

Table A
\begin{tabular}{|l|c|c|c|}
\hline
[tex]$x$[/tex] & 0.75 & 1 & 1.25 \\
\hline
[tex]$a(x)$[/tex] & 1.03818 & 1.05121 & 1.06439 \\
\hline
\end{tabular}

Table B
\begin{tabular}{|c|c|c|c|}
\hline
[tex]$d$[/tex] & [tex]$1{,}038.18$[/tex] & [tex]$1{,}051.21$[/tex] & [tex]$1{,}064.39$[/tex] \\
\hline
[tex]$r(d)$[/tex] & 0.75 & 1 & 1.25 \\
\hline
\end{tabular}

Use the values in the tables 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 solve this problem, we need to verify the relationship between the given functions in Table A and Table B.

### Step-by-Step Verification

1. Understand the Tables:
- Table A:
- Represents the function \( a(x) \) which models the total amount of the investment based on the annual interest rate \( x \% \).
- \( x \) values: \( 0.75\%, 1\%, 1.25\% \)
- \( a(x) \) values: \( 1.03818, 1.05121, 1.06439 \) (which are likely meant to be \( 1038.18, 1051.21, 1064.39 \), but we will follow the given numbers)

- Table B:
- Represents the function \( r(d) \) which models the interest rate based on the total amount at the end of the investment, \( d \).
- \( d \) values: \( 1057.81, 1077.78, 1098.12 \)
- \( r(d) \) values: \( 0.75, 1, 1.25 \)

2. Check for Inverses:
- If two functions are inverses of each other, then for each pair \((x, y)\) in one function, there should be a corresponding pair \((y, x)\) in the other function.

3. Verification:

- Compare domain and range of Table A and Table B:
- Domain of Table A: \( \{0.75, 1, 1.25\} \)
- Range of Table B (i.e., \( r(d) \)): \( \{0.75, 1, 1.25\} \)
- Since the domain of Table A is the same as the range of Table B, one part of the inverse function condition is satisfied.

- Compare range of Table A and domain of Table B:
- Range of Table A (i.e., \( a(x) \)): \( \{1.03818, 1.05121, 1.06439\} \)
- Domain of Table B: \( \{1057.81, 1077.78, 1098.12 \} \)
- The range of Table A and the domain of Table B are different.

4. Conclusions Based on the Verification:
- The functions cannot be inverses of each other because for each ordered pair \((x, y)\) in one function, there is no corresponding ordered pair \((y, x)\) in the other function – this condition applies and disqualifies the functions as true inverses.
- However, since the domain of Table A is the same as the range of Table B, this implies some level of functional relationship, but it does not fully satisfy the requirements of inverse functions.

### Conclusion
Given the specific numerical verification and comparisons:

- The functions are inverses because the domain of Table A is the same as the range of Table B.

Therefore, the detailed solution verifies that the most appropriate conclusion is:

"The functions are inverses because the domain of Table [tex]\(A\)[/tex] is the same as the range of Table [tex]\(B\)[/tex]."

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