Answer :
To determine which function represents the resale value of a textbook that decreases by 25% with each previous owner, let’s break it down step by step.
1. Initial Value (New Textbook Price):
- The price of a new textbook is $85.
2. Decrease in Value:
- The textbook decreases in value by 25% with each previous owner.
- Decreasing by 25% means retaining 75% of its value, because [tex]\(100\% - 25\% = 75\%\)[/tex].
- Mathematically, retaining 75% of the value is the same as multiplying by 0.75.
3. Value After Each Owner:
- After the first previous owner, the value of the textbook would be [tex]\(85 \times 0.75\)[/tex].
- After the second previous owner, the value would be [tex]\(85 \times 0.75 \times 0.75\)[/tex].
- After the third previous owner, the value would be [tex]\(85 \times 0.75 \times 0.75 \times 0.75\)[/tex].
- Generally, after [tex]\(x\)[/tex] owners, the value would be [tex]\(85 \times 0.75^x\)[/tex].
4. Function Representation:
- We can see that the value is decreasing exponentially based on the number of owners.
- Therefore, the function that represents this pattern is:
[tex]\[ f(x) = 85 \times (0.75)^x \][/tex]
5. Matching the Given Options:
- Among the options, the correct one is:
[tex]\[ f(x) = 85 \times (1 - 0.25)^x \][/tex]
- This is equivalent to [tex]\(85 \times 0.75^x\)[/tex] because [tex]\(1 - 0.25 = 0.75\)[/tex].
Therefore, the correct answer is:
[tex]\[ f(x) = 85(1 - 0.25)^x \][/tex]
This validates that the function chosen correctly models the decrease in the resale value of the textbook with each owner.
1. Initial Value (New Textbook Price):
- The price of a new textbook is $85.
2. Decrease in Value:
- The textbook decreases in value by 25% with each previous owner.
- Decreasing by 25% means retaining 75% of its value, because [tex]\(100\% - 25\% = 75\%\)[/tex].
- Mathematically, retaining 75% of the value is the same as multiplying by 0.75.
3. Value After Each Owner:
- After the first previous owner, the value of the textbook would be [tex]\(85 \times 0.75\)[/tex].
- After the second previous owner, the value would be [tex]\(85 \times 0.75 \times 0.75\)[/tex].
- After the third previous owner, the value would be [tex]\(85 \times 0.75 \times 0.75 \times 0.75\)[/tex].
- Generally, after [tex]\(x\)[/tex] owners, the value would be [tex]\(85 \times 0.75^x\)[/tex].
4. Function Representation:
- We can see that the value is decreasing exponentially based on the number of owners.
- Therefore, the function that represents this pattern is:
[tex]\[ f(x) = 85 \times (0.75)^x \][/tex]
5. Matching the Given Options:
- Among the options, the correct one is:
[tex]\[ f(x) = 85 \times (1 - 0.25)^x \][/tex]
- This is equivalent to [tex]\(85 \times 0.75^x\)[/tex] because [tex]\(1 - 0.25 = 0.75\)[/tex].
Therefore, the correct answer is:
[tex]\[ f(x) = 85(1 - 0.25)^x \][/tex]
This validates that the function chosen correctly models the decrease in the resale value of the textbook with each owner.
