Question

Trent purchased a house that was worth $191,000. The value of the house increased by 9% each year for the next 5 years. Write a function f(x) that determines the value of the house (in thousands of dollars) in terms of the number of years t since Trent purchased the house.

Answer 1

f(t) = 191 *
`(1.09)^t` in thousands of dollars.

Step-by-step explanation:

The function f(t) = 191 * (1.09)^t models the value of Trent's house in thousands of dollars after t years. The initial value of the house, $191,000, is multiplied by the growth factor of 1.09, representing a 9% annual increase.

The exponent 't' signifies the number of years since the house was purchased, influencing the compound growth. As time progresses (t increases), the function calculates the value of the house by compounding the 9% annual increase.

This exponential growth formula captures the cumulative effect of the annual 9% increment on the original value over time. Hence, the function allows us to determine the value of the house at any point in the future based on the number of years since Trent purchased it.

Answer 2

This function
`\( f(t) \)` will give you the value of the house in thousands of dollars after
`\( t \)` years.

To write a function \( f(t) \) that determines the value of Trent's house (in thousands of dollars) in terms of the number of years \( t \) since he purchased the house, we need to account for an annual increase of 9% in the house's value. The function will be an exponential growth function because the house value increases by a constant percentage each year.

The general form of an exponential growth function is:

`\[ f(t) = P * (1 + r)^t \]`

where:

-
`\( P \)` is the initial amount (the initial value of the house).

-
`\( r \)` is the rate of increase (9% in this case, which is 0.09 as a decimal).

-
`\( t \)` is the time in years.

-
`\( f(t) \)` is the value after
`\( t \)` years.

Given that the initial value of the house is $191,000, but the value is requested in thousands of dollars,
`\( P = 191 \)` (since $191,000 is equivalent to 191 thousand dollars).

Thus, the function
`\( f(t) \)` is:

`\[ f(t) = 191 * (1 + 0.09)^t \]`

`\[ f(t) = 191 * 1.09^t \]`

This function
`\( f(t) \)` will give you the value of the house in thousands of dollars after
`\( t \)` years.