Alright, let's break this down step-by-step to understand which function correctly represents the resale value of the textbook after [tex]\(x\)[/tex] owners.
1. Initial Value of the Textbook:
- We are given that a new textbook sells for [tex]$85.
- Therefore, the initial value (before any depreciation) is $[/tex]85.
2. Depreciation Rate:
- The value of the textbook decreases by [tex]\(25\%\)[/tex] with each previous owner.
- Depreciation rate of [tex]\(25\%\)[/tex] can be expressed as a decimal: [tex]\(0.25\)[/tex].
3. Understanding Depreciation:
- Each time the textbook is resold, it retains [tex]\(100\% - 25\% = 75\% \)[/tex] of its previous value.
- In decimal form, each resale keeps [tex]\(0.75\)[/tex] (or [tex]\(1 - 0.25\)[/tex]) of the previous value.
4. Formulating the Function:
- The resale value after [tex]\(1\)[/tex] owner would be:
[tex]\[
85 \times 0.75
\][/tex]
- After [tex]\(2\)[/tex] owners, the resale value would be:
[tex]\[
85 \times 0.75 \times 0.75 = 85 \times 0.75^2
\][/tex]
- After [tex]\(3\)[/tex] owners:
[tex]\[
85 \times 0.75 \times 0.75 \times 0.75 = 85 \times 0.75^3
\][/tex]
- Generalizing this for [tex]\(x\)[/tex] owners, the resale value would be:
[tex]\[
85 \times 0.75^x
\][/tex]
5. Writing the Function in Standard Form:
- The formula can be expressed as:
[tex]\[
f(x) = 85 \times (1 - 0.25)^x
\][/tex]
or equivalently:
[tex]\[
f(x) = 85 \times 0.75^x
\][/tex]
6. Choosing the Correct Function:
- Examining the provided options:
- [tex]\(f(x)=85(1-0.25)^x\)[/tex]
- [tex]\(f(x)=85(1+0.25)^x\)[/tex]
- [tex]\(f(x)=85(0.25)^x\)[/tex]
- [tex]\(f(x)=(85-0.25)^x\)[/tex]
- The correct function that represents the resale value of the textbook after [tex]\(x\)[/tex] owners is:
[tex]\[
f(x) = 85(1-0.25)^x
\][/tex]
Hence, the function that accurately represents the resale value of the textbook after [tex]\(x\)[/tex] owners is:
[tex]\[
\boxed{f(x)=85(1-0.25)^x}
\][/tex]
