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Sagot :
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]."
### 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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