To determine the function that can be used to find the value of a home after \( x \) years, given that its value increases at an average rate of 5.5% each year and its current value is $120,000, we need to understand the principles of exponential growth.
Here's a detailed, step-by-step solution:
1. Understanding the growth rate:
- The growth rate is given as 5.5% per year.
- To convert that to a multiplier, we add 1 to the rate:
[tex]\[
\text{Multiplier} = 1 + \frac{5.5}{100} = 1 + 0.055 = 1.055
\][/tex]
2. Defining the exponential growth function:
- The home value increases by a multiplicative factor of 1.055 each year.
- Hence, after \( x \) years, the home value will be multiplied by \( (1.055)^x \).
3. Establishing the function:
- The current value of the home is $120,000.
- The function representing the home value after \( x \) years will be the current value multiplied by the growth factor raised to the power \( x \):
[tex]\[
f(x) = 120,000 \times (1.055)^x
\][/tex]
Given the choices, the correct function that represents the value of the home after \( x \) years is:
[tex]\[
f(x) = 120,000 (1.055)^x
\][/tex]
Thus, the correct answer is:
[tex]\[
f(x) = 120,000 (1.055)^x
\][/tex]
