To determine the correct equation representing the value of the home [tex]\( x \)[/tex] years after the purchase, we need to consider the given details:
1. The initial value of the home is \[tex]$230,000.
2. The home value decreases by 2% each year.
Let \( f(x) \) represent the value of the home \( x \) years after the purchase.
### Step-by-Step Solution
1. Initial Value:
The initial value of the home when \( x = 0 \) is \$[/tex]230,000. This is represented by [tex]\( 230000 \)[/tex].
2. Annual Decrease Factor:
Since the value decreases by 2% each year, the remaining value each year is [tex]\( 100\% - 2\% = 98\% \)[/tex].
3. Decimal Representation:
To use this in the equation, we convert 98% to a decimal, which is [tex]\( 0.98 \)[/tex].
4. Exponential Decay Representation:
The value of the home decreases exponentially each year by a factor of [tex]\( 0.98 \)[/tex]. Therefore, after [tex]\( x \)[/tex] years, the value of the home can be represented as:
[tex]\[
f(x) = 230000 \times (0.98)^x
\][/tex]
### Conclusion
The correct equation that can be used to represent the value of the home [tex]\( x \)[/tex] years after the purchase is:
[tex]\[
f(x) = 230000 (0.98)^x
\][/tex]
### Comparison with Other Options
Let's compare this with the options given:
- [tex]\( f(x)=230000(0.98)^x \)[/tex]: This matches our derived equation.
- [tex]\( f(x)=230000(1.02)^x \)[/tex]: This represents a 2% increase per year, not a 2% decrease.
- [tex]\( f(x)=2(0.98)^x \)[/tex]: This does not account for the initial value of \$230,000.
- [tex]\( f(x)=2(1.02)^x \)[/tex]: This represents a 2% increase per year with an incorrect initial value.
Therefore, the correct answer is:
[tex]\[ f(x)=230000(0.98)^x \][/tex]
