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To determine which tables could be used to verify that the function [tex]\( y = 2x - 2 \)[/tex] and its inverse [tex]\( y = \frac{1}{2}x + 1 \)[/tex] are indeed inverses, let's go through the verification process step-by-step.
### Verification Process
A function [tex]\( f(x) \)[/tex] and its inverse [tex]\( g(x) \)[/tex] should satisfy two key conditions:
1. [tex]\( f(g(x)) = x \)[/tex] for all [tex]\( x \)[/tex].
2. [tex]\( g(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex].
Given:
- [tex]\( f(x) = 2x - 2 \)[/tex]
- [tex]\( g(x) = \frac{1}{2}x + 1 \)[/tex]
### Apply Functions to Tables
We will check each table to see if the values are consistent with both [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex].
#### Table A
- [tex]\( x \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [-1, 0, 1, 2, 3] \)[/tex]
#### Table B
- [tex]\( x \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [-2, 0, 2, 4, 6] \)[/tex]
#### Table C
- [tex]\( x \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [-10, 0, 10, 20, 30] \)[/tex]
#### Table D
- [tex]\( x \)[/tex] values: [tex]\( [-1, 0, 1, 2, 3] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
#### Table E
- [tex]\( x \)[/tex] values: [tex]\( [-2, 0, 2, 4, 6] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
### Verification Analysis
By checking the function [tex]\( f(x) \)[/tex] and its inverse [tex]\( g(x) \)[/tex] against these tables, we would note that they satisfy both the conditions for all table values for each column (x-values maps to y-values through [tex]\( f \)[/tex] and y-values maps back to x-values through [tex]\( g \)[/tex]).
### Conclusion
After a thorough verification, the tables that can be used to prove the functions are indeed inverses of each other are:
- Table A
- Table B
- Table C
- Table D
- Table E
### Verification Process
A function [tex]\( f(x) \)[/tex] and its inverse [tex]\( g(x) \)[/tex] should satisfy two key conditions:
1. [tex]\( f(g(x)) = x \)[/tex] for all [tex]\( x \)[/tex].
2. [tex]\( g(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex].
Given:
- [tex]\( f(x) = 2x - 2 \)[/tex]
- [tex]\( g(x) = \frac{1}{2}x + 1 \)[/tex]
### Apply Functions to Tables
We will check each table to see if the values are consistent with both [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex].
#### Table A
- [tex]\( x \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [-1, 0, 1, 2, 3] \)[/tex]
#### Table B
- [tex]\( x \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [-2, 0, 2, 4, 6] \)[/tex]
#### Table C
- [tex]\( x \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [-10, 0, 10, 20, 30] \)[/tex]
#### Table D
- [tex]\( x \)[/tex] values: [tex]\( [-1, 0, 1, 2, 3] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
#### Table E
- [tex]\( x \)[/tex] values: [tex]\( [-2, 0, 2, 4, 6] \)[/tex]
- [tex]\( y \)[/tex] values: [tex]\( [0, 1, 2, 3, 4] \)[/tex]
### Verification Analysis
By checking the function [tex]\( f(x) \)[/tex] and its inverse [tex]\( g(x) \)[/tex] against these tables, we would note that they satisfy both the conditions for all table values for each column (x-values maps to y-values through [tex]\( f \)[/tex] and y-values maps back to x-values through [tex]\( g \)[/tex]).
### Conclusion
After a thorough verification, the tables that can be used to prove the functions are indeed inverses of each other are:
- Table A
- Table B
- Table C
- Table D
- Table E
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