Answer :
Let's analyze the given function and fill in the blanks with interpretations and appropriate values.
### Given Function:
[tex]\[ f(x) = \frac{1}{2} x (12 - 2x) \][/tex]
### Calculations:
1. Calculate [tex]\( f(-3) \)[/tex]:
[tex]\[ f(-3) = \frac{1}{2} \times (-3) \times (12 - 2 \times (-3)) = \frac{1}{2} \times (-3) \times (12 + 6) = \frac{1}{2} \times (-3) \times 18 = (-3) \times 9 = -27 \][/tex]
2. Calculate [tex]\( f(2.5) \)[/tex]:
[tex]\[ f(2.5) = \frac{1}{2} \times 2.5 \times (12 - 2 \times 2.5) = \frac{1}{2} \times 2.5 \times (12 - 5) = \frac{1}{2} \times 2.5 \times 7 = 1.25 \times 7 = 8.75 \][/tex]
3. Calculate [tex]\( f(7) \)[/tex]:
[tex]\[ f(7) = \frac{1}{2} \times 7 \times (12 - 2 \times 7) = \frac{1}{2} \times 7 \times (12 - 14) = \frac{1}{2} \times 7 \times (-2) = 7 \times (-1) = -7 \][/tex]
### Interpretation of Function Values:
- [tex]\( f(-3) = -27 \)[/tex]:
This means when the width of the rectangular area is [tex]\(-3\)[/tex] feet, the area would be [tex]\(-27\)[/tex] square feet. This interpretation does not make sense in the context of the problem since a rectangle cannot have a negative width.
- [tex]\( f(2.5) = 8.75 \)[/tex]:
This means when the width of the rectangular area is [tex]\(2.5\)[/tex] feet, the area would be [tex]\(8.75\)[/tex] square feet. This interpretation is valid in the context of the problem.
- [tex]\( f(7) = -7 \)[/tex]:
This means when the width of the rectangular area is [tex]\(7\)[/tex] feet, the area would be [tex]\(-7\)[/tex] square feet. This interpretation does not make sense in the context of the problem because, given 12 feet of fencing, the length (which comes out to be negative in this case) cannot logically support this dimension.
### Determining an Appropriate Domain:
To determine an appropriate domain, we consider the physical constraints of the problem where both width and length must be non-negative.
- For a valid rectangular area, the width [tex]\(x\)[/tex] must be positive and less than half the total available fencing since the width and length both use up the fencing.
- [tex]\( 0 < x < 6 \)[/tex]:
This is because, when [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = \frac{1}{2} \times 6 \times (12 - 2 \times 6) = \frac{1}{2} \times 6 \times 0 = 0 \][/tex]
Thus, the appropriate domain for the function is [tex]\( 0 < x < 6 \)[/tex].
Based on the information above, here are the completed answers with interpretations and the appropriate domain:
[tex]\[ f(-3) = -27, \text{ meaning when the width of the rectangular area is } -3 \text{ ft, the area would be } -27 \text{ ft}^2. \text{ This interpretation does not make sense in the context of the problem.} \][/tex]
[tex]\[ f(2.5) = 8.75, \text{ meaning when the width of the rectangular area is } 2.5 \text{ ft, the area would be 8.75 ft}^2. \text{ This interpretation makes sense in the context of the problem.} \][/tex]
[tex]\[ f(7) = -7, \text{ meaning when the width of the rectangular area is } 7 \text{ ft, the area would be } -7 \text{ ft}^2. \text{ This interpretation does not make sense in the context of the problem.} \][/tex]
Based on the observations above, it is clear that an appropriate domain for the function is [tex]\(0 < x < 6\)[/tex].
### Given Function:
[tex]\[ f(x) = \frac{1}{2} x (12 - 2x) \][/tex]
### Calculations:
1. Calculate [tex]\( f(-3) \)[/tex]:
[tex]\[ f(-3) = \frac{1}{2} \times (-3) \times (12 - 2 \times (-3)) = \frac{1}{2} \times (-3) \times (12 + 6) = \frac{1}{2} \times (-3) \times 18 = (-3) \times 9 = -27 \][/tex]
2. Calculate [tex]\( f(2.5) \)[/tex]:
[tex]\[ f(2.5) = \frac{1}{2} \times 2.5 \times (12 - 2 \times 2.5) = \frac{1}{2} \times 2.5 \times (12 - 5) = \frac{1}{2} \times 2.5 \times 7 = 1.25 \times 7 = 8.75 \][/tex]
3. Calculate [tex]\( f(7) \)[/tex]:
[tex]\[ f(7) = \frac{1}{2} \times 7 \times (12 - 2 \times 7) = \frac{1}{2} \times 7 \times (12 - 14) = \frac{1}{2} \times 7 \times (-2) = 7 \times (-1) = -7 \][/tex]
### Interpretation of Function Values:
- [tex]\( f(-3) = -27 \)[/tex]:
This means when the width of the rectangular area is [tex]\(-3\)[/tex] feet, the area would be [tex]\(-27\)[/tex] square feet. This interpretation does not make sense in the context of the problem since a rectangle cannot have a negative width.
- [tex]\( f(2.5) = 8.75 \)[/tex]:
This means when the width of the rectangular area is [tex]\(2.5\)[/tex] feet, the area would be [tex]\(8.75\)[/tex] square feet. This interpretation is valid in the context of the problem.
- [tex]\( f(7) = -7 \)[/tex]:
This means when the width of the rectangular area is [tex]\(7\)[/tex] feet, the area would be [tex]\(-7\)[/tex] square feet. This interpretation does not make sense in the context of the problem because, given 12 feet of fencing, the length (which comes out to be negative in this case) cannot logically support this dimension.
### Determining an Appropriate Domain:
To determine an appropriate domain, we consider the physical constraints of the problem where both width and length must be non-negative.
- For a valid rectangular area, the width [tex]\(x\)[/tex] must be positive and less than half the total available fencing since the width and length both use up the fencing.
- [tex]\( 0 < x < 6 \)[/tex]:
This is because, when [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = \frac{1}{2} \times 6 \times (12 - 2 \times 6) = \frac{1}{2} \times 6 \times 0 = 0 \][/tex]
Thus, the appropriate domain for the function is [tex]\( 0 < x < 6 \)[/tex].
Based on the information above, here are the completed answers with interpretations and the appropriate domain:
[tex]\[ f(-3) = -27, \text{ meaning when the width of the rectangular area is } -3 \text{ ft, the area would be } -27 \text{ ft}^2. \text{ This interpretation does not make sense in the context of the problem.} \][/tex]
[tex]\[ f(2.5) = 8.75, \text{ meaning when the width of the rectangular area is } 2.5 \text{ ft, the area would be 8.75 ft}^2. \text{ This interpretation makes sense in the context of the problem.} \][/tex]
[tex]\[ f(7) = -7, \text{ meaning when the width of the rectangular area is } 7 \text{ ft, the area would be } -7 \text{ ft}^2. \text{ This interpretation does not make sense in the context of the problem.} \][/tex]
Based on the observations above, it is clear that an appropriate domain for the function is [tex]\(0 < x < 6\)[/tex].
