Sure, let’s break down the process to find the function that represents the car’s value after [tex]\( x \)[/tex] years, given that the car’s value depreciates by 15% each year.
1. Initial Condition: Terrence buys a car for [tex]$20,000.
2. Depreciation Rate: Each year, the value of the car decreases by 15%.
3. Depreciation Factor: To understand how depreciation works mathematically, if the car depreciates by 15%, it retains 85% (100% - 15% = 85% or 0.85) of its value each year.
4. General Formula: The value of the car after one year would be \( \$[/tex]20,000 \times 0.85 \). If you want to find the value after two years, it would be [tex]\( \$20,000 \times 0.85 \times 0.85 \)[/tex] or [tex]\( \$20,000 \times (0.85)^2 \)[/tex]. This pattern continues:
After [tex]\( x \)[/tex] years, the value of the car would be [tex]\( \$20,000 \times (0.85)^x \)[/tex].
5. Conclusion: Hence, the function [tex]\( f(x) \)[/tex] that represents the car’s value after [tex]\( x \)[/tex] years is:
[tex]\[
f(x) = 20,000 \times (0.85)^x
\][/tex]
So, the correct function is:
[tex]\[ f(x) = 20,000(0.85)^x \][/tex]
Thus, the best choice among the given options is:
[tex]\[ \boxed{f(x) = 20,000(0.85)^x} \][/tex]
