Certainly! Let's go through the problem step-by-step.
1. Initial Purchase Value:
- Terrence buys a new car for \[tex]$20,000.
2. Depreciation Rate:
- The value of the car depreciates by 15% every year.
3. Understand Depreciation:
- Depreciation means a decrease in value.
- A 15% depreciation each year means that each year the car retains 85% of its value. This is calculated as \(100\% - 15\% = 85\%\).
4. Express Remaining Value as a Decimal:
- 85% in decimal form is 0.85.
5. Setting Up the Depreciation Function:
- Let \(f(x)\) represent the value of the car after \(x\) years.
- Initially, at \(x = 0\) years, the value is \$[/tex]20,000.
- After 1 year, the value is [tex]\(20,000 \times 0.85\)[/tex].
- After 2 years, the value is [tex]\(20,000 \times 0.85 \times 0.85 = 20,000 \times (0.85)^2\)[/tex].
- After 3 years, the value is [tex]\(20,000 \times 0.85 \times 0.85 \times 0.85 = 20,000 \times (0.85)^3\)[/tex].
- Generally, after [tex]\(x\)[/tex] years, the value will be [tex]\(20,000 \times (0.85)^x\)[/tex].
6. Conclusion:
- The function [tex]\(f(x)\)[/tex] that represents the value of the car after [tex]\(x\)[/tex] years is:
[tex]\[
f(x) = 20,000 \times (0.85)^x
\][/tex]
Among the given options:
- [tex]\(f(x) = 20,000(0.85)^x\)[/tex]
- [tex]\(f(x) = 20,000(0.15)^x\)[/tex]
- [tex]\(f(x) = 20,000(1.15)^x\)[/tex]
- [tex]\(f(x) = 20,000(1.85)^x\)[/tex]
The correct function is:
[tex]\[
f(x) = 20,000(0.85)^x
\][/tex]
