A business purchases a computer for $$ \$ 3,000 $$. The value of the computer decreases at a rate of $$ 15 \% $$ per year. Write an exponential function to model this situation. Then determine how much the computer will be worth after 4 years. function $$ f(x)=\square $$ value after 4 years: $$ \$ \square $$

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To model the depreciation of the computer's value over time, we can use an exponential decay function. Here's how to set it up and calculate the value after 4 years:

### Exponential Function Model

The general form of an exponential decay function is:

$$ f(x) = a \times (1 - r)^x $$

Where:
- $$ a $$ is the initial value (purchase price).
- $$ r $$ is the annual depreciation rate.
- $$ x $$ is the number of years.

Given:
- **Initial Value ($$ a $$)**: \$3,000
- **Depreciation Rate ($$ r $$)**: 15% or 0.15

Plugging these values into the formula:

$$ f(x) = 3000 \times (1 - 0.15)^x $$
$$ f(x) = 3000 \times (0.85)^x $$

### Value After 4 Years

To find the value of the computer after 4 years ($$ x = 4 $$):

$$ f(4) = 3000 \times (0.85)^4 $$

Calculating $$ (0.85)^4 $$:

$$ (0.85)^4 \approx 0.522 $$

Now, multiply by the initial value:

$$ f(4) = 3000 \times 0.522 $$
$$ f(4) \approx 1,566 $$

### Summary

- **Function**:
  
  $$ f(x) = 3000 \times (0.85)^x $$

- **Value After 4 Years**:
  
  \$1,566
Function: $$ f(x) = 3000 \times (0.85)^x $$ Value after 4 years: \$1,566

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