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Consider the tables created using an initial investment of \$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}{|c|c|c|c|}
\hline
[tex]x[/tex] & 0.75 & 1 & 1.25 \\
\hline
[tex]a(x)[/tex] & \$1,038.18 & \[tex]$1,051.21 & \$[/tex]1,064.39 \\
\hline
\end{tabular}

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

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.

Sagot :

GinnyAnsweridn
To determine the relationship between the functions represented in Table A and Table B, we need to assess whether each function is the inverse of the other. The concept of inverse functions means that if we have a function [tex]\( f(a) = b \)[/tex], then its inverse [tex]\( f^{-1}(b) = a \)[/tex].

Let's start by examining Table A and Table B:

### Table A
| [tex]\( x \)[/tex] | [tex]\( a(x) \)[/tex] |
|---------|--------------|
| 0.75 | 1038.18 |
| 1 | 1051.21 |
| 1.25 | 1064.39 |

### Table B
| [tex]\( d \)[/tex] | [tex]\( r(d) \)[/tex] |
|-------------|------------|
| 1057.81 | 0.75 |
| 1077.78 | 1 |
| 1098.12 | 1.25 |

To verify if these functions are inverses, we need to check if the output of Table A for a given input is the same as the input of Table B for the corresponding output, and vice versa.

1. Check if [tex]\( a(x) \)[/tex] in Table A matches [tex]\( d \)[/tex] and [tex]\( r(d) = x \)[/tex] in Table B:

- From Table A, when [tex]\( x = 0.75, a(x) = 1038.18 \)[/tex]:
- We should have in Table B: [tex]\( r(1038.18) \)[/tex] equals [tex]\( 0.75 \)[/tex] with [tex]\( d \)[/tex] that might match [tex]\( 1038.18 \)[/tex]
- However, Table B does not include [tex]\( 1038.18 \)[/tex], so the required entry is missing.

- From Table A, when [tex]\( x = 1, a(x) = 1051.21 \)[/tex]:
- We should have in Table B: [tex]\( r(1051.21) \)[/tex] equals [tex]\( 1 \)[/tex] with [tex]\( d \)[/tex] that might match [tex]\( 1051.21 \)[/tex]
- Again, Table B does not include [tex]\( 1051.21 \)[/tex], so the required entry is missing.

- From Table A, when [tex]\( x = 1.25, a(x) = 1064.39 \)[/tex]:
- We should have in Table B: [tex]\( r(1064.39) \)[/tex] equals [tex]\( 1.25 \)[/tex] with [tex]\( d \)[/tex] that might match [tex]\( 1064.39 \)[/tex]
- Table B does not include [tex]\( 1064.39 \)[/tex], so the required entry is missing.

2. Check if [tex]\( r(d) \)[/tex] in Table B matches [tex]\( x \)[/tex] and [tex]\( a(x) = d \)[/tex] in Table A:

- From Table B, when [tex]\( d = 1057.81, r(d) = 0.75 \)[/tex]:
- We should have in Table A: [tex]\( a(0.75) \)[/tex] equals [tex]\( 1057.81 \)[/tex]
- However, in Table A, [tex]\( a(0.75) = 1038.18 \)[/tex], not [tex]\( 1057.81 \)[/tex]

- From Table B, when [tex]\( d = 1077.78, r(d) = 1 \)[/tex]:
- We should have in Table A: [tex]\( a(1) \)[/tex] equals [tex]\( 1077.78 \)[/tex]
- But in Table A, [tex]\( a(1) = 1051.21 \)[/tex], not [tex]\( 1077.78 \)[/tex]

- From Table B, when [tex]\( d = 1098.12, r(d) = 1.25 \)[/tex]:
- We should have in Table A: [tex]\( a(1.25) \)[/tex] equals [tex]\( 1098.12 \)[/tex]
- But in Table A, [tex]\( a(1.25) = 1064.39 \)[/tex], not [tex]\( 1098.12 \)[/tex]

As observed, the corresponding values in these tables do not match in a way that would confirm the functions as inverses. Thus, considering these observations, we conclude:

_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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