Answer :
Given the function [tex]\( f(x) = 2x + 2 \)[/tex], we need to find and interpret the function values for specific inputs, as well as determine an appropriate domain for the function.
1. Calculation and Interpretation of [tex]\( f(2) \)[/tex]:
- Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 2 \cdot 2 + 2 = 4 + 2 = 6 \][/tex]
- Interpretation:
[tex]\[ f(2) = 6 \][/tex]
Therefore, when a 2 is rolled on the die, the player is awarded 6 points. This interpretation is valid in the context of the problem since a die roll of 2 is within the range of possible die outcomes (1 to 6).
2. Calculation and Interpretation of [tex]\( f(3.5) \)[/tex]:
- Calculate [tex]\( f(3.5) \)[/tex]:
[tex]\[ f(3.5) = 2 \cdot 3.5 + 2 = 7 + 2 = 9.0 \][/tex]
- Interpretation:
[tex]\[ f(3.5) = 9.0 \][/tex]
Hence, when a 3.5 is rolled on the die, the player is awarded 9.0 points. However, rolling a 3.5 is not realistic with a standard six-sided die, indicating this interpretation is not entirely valid within the physical constraints of the game.
3. Calculation and Interpretation of [tex]\( f(7) \)[/tex]:
- Calculate [tex]\( f(7) \)[/tex]:
[tex]\[ f(7) = 2 \cdot 7 + 2 = 14 + 2 = 16 \][/tex]
- Interpretation:
[tex]\[ f(7) = 16 \][/tex]
Thus, when a 7 is rolled on the die, the player is awarded 16 points. Similar to the previous case, rolling a 7 on a standard six-sided die is impossible, making this interpretation also invalid in the context of a standard die.
4. Determine the Appropriate Domain:
- Based on the context of using a six-sided die, the only valid inputs for the function [tex]\( f \)[/tex] are the integer values from 1 to 6, inclusive. This is because these are the possible outcomes when rolling a standard die.
Therefore, the appropriate domain for the function is:
[tex]\[ [1, 2, 3, 4, 5, 6] \][/tex]
So the complete, detailed answer is as follows:
1. [tex]\( f(2) = 6 \)[/tex], meaning when a 2 is rolled on the die, the player is awarded 6 points. This interpretation is valid in the context of the problem.
2. [tex]\( f(3.5) = 9.0 \)[/tex], meaning when a 3.5 is rolled on the die, the player is awarded 9.0 points. This interpretation is not valid in the context of the problem as 3.5 cannot be rolled on a standard six-sided die.
3. [tex]\( f(7) = 16 \)[/tex], meaning when a 7 is rolled on the die, the player is awarded 16 points. This interpretation is not valid in the context of the problem as 7 cannot be rolled on a standard six-sided die.
Based on the observations above, it is clear that an appropriate domain for the function is:
[tex]\[ [1, 2, 3, 4, 5, 6] \][/tex]
1. Calculation and Interpretation of [tex]\( f(2) \)[/tex]:
- Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 2 \cdot 2 + 2 = 4 + 2 = 6 \][/tex]
- Interpretation:
[tex]\[ f(2) = 6 \][/tex]
Therefore, when a 2 is rolled on the die, the player is awarded 6 points. This interpretation is valid in the context of the problem since a die roll of 2 is within the range of possible die outcomes (1 to 6).
2. Calculation and Interpretation of [tex]\( f(3.5) \)[/tex]:
- Calculate [tex]\( f(3.5) \)[/tex]:
[tex]\[ f(3.5) = 2 \cdot 3.5 + 2 = 7 + 2 = 9.0 \][/tex]
- Interpretation:
[tex]\[ f(3.5) = 9.0 \][/tex]
Hence, when a 3.5 is rolled on the die, the player is awarded 9.0 points. However, rolling a 3.5 is not realistic with a standard six-sided die, indicating this interpretation is not entirely valid within the physical constraints of the game.
3. Calculation and Interpretation of [tex]\( f(7) \)[/tex]:
- Calculate [tex]\( f(7) \)[/tex]:
[tex]\[ f(7) = 2 \cdot 7 + 2 = 14 + 2 = 16 \][/tex]
- Interpretation:
[tex]\[ f(7) = 16 \][/tex]
Thus, when a 7 is rolled on the die, the player is awarded 16 points. Similar to the previous case, rolling a 7 on a standard six-sided die is impossible, making this interpretation also invalid in the context of a standard die.
4. Determine the Appropriate Domain:
- Based on the context of using a six-sided die, the only valid inputs for the function [tex]\( f \)[/tex] are the integer values from 1 to 6, inclusive. This is because these are the possible outcomes when rolling a standard die.
Therefore, the appropriate domain for the function is:
[tex]\[ [1, 2, 3, 4, 5, 6] \][/tex]
So the complete, detailed answer is as follows:
1. [tex]\( f(2) = 6 \)[/tex], meaning when a 2 is rolled on the die, the player is awarded 6 points. This interpretation is valid in the context of the problem.
2. [tex]\( f(3.5) = 9.0 \)[/tex], meaning when a 3.5 is rolled on the die, the player is awarded 9.0 points. This interpretation is not valid in the context of the problem as 3.5 cannot be rolled on a standard six-sided die.
3. [tex]\( f(7) = 16 \)[/tex], meaning when a 7 is rolled on the die, the player is awarded 16 points. This interpretation is not valid in the context of the problem as 7 cannot be rolled on a standard six-sided die.
Based on the observations above, it is clear that an appropriate domain for the function is:
[tex]\[ [1, 2, 3, 4, 5, 6] \][/tex]
