(a) The linear function is \(V( x) = - 300x + 1500\).

(b) The implied domain is \([ 0, 5] \).

Free Solution By Steps from UpStudy

Step 1: Identify the initial value and depreciation rate

- The initial value of the computer is $1500.

- The computer depreciates to $0 over 5 years, which implies a straight-line depreciation.

Step 2: Calculate the annual depreciation rate

- The total depreciation over 5 years is $1500.

- Annual depreciation rate = Total depreciation / Number of years = $1500 / 5 = $300 per year.

Step 3: Write the linear depreciation function

- The function can be written in slope-intercept form \(V( x) = mx + b\), where \(m\) is the slope (depreciation rate) and \(b\) is the initial value.

- Here, \(m = - 300\) (since the value decreases) and \(b = 1500\).

So, the function is:

\[V( x) = - 300x + 1500\]

Step 4: Determine the implied domain

- The domain of the function represents the range of ages from 0 to 5 years, inclusive, because the computer depreciates to $0 over this period.

- In interval notation, the domain is \([ 0, 5] \).

Key Concepts:

1. Linear Depreciation: Linear depreciation refers to the method where an asset loses value at a constant rate over time. This is often represented by a linear function.

2. Slope-Intercept Form: A linear function can be written in the form \(V( x) = mx + b\), where \(m\) is the slope, representing the rate of change, and \(b\) is the y-intercept, representing the initial value.

Explanation:

- Initial Value and Depreciation Rate: The initial value is the starting value of the asset, and the depreciation rate is the amount by which the asset's value decreases each year. For instance, if a car is worth $20,000 initially and loses $4,000 per year, the depreciation rate is $4,000.

- Annual Depreciation Calculation: To find the annual depreciation rate, divide the total depreciation by the number of years. For example, if a machine worth $10,000 depreciates to $0 over 5 years, the annual depreciation is:

\[\text { Annual Depreciation} = \frac { 10,000} { 5} = 2,000\]

- Writing the Linear Depreciation Function: The depreciation function describes how the value of the asset changes over time. The slope \( m \) is negative since the value decreases. For example, if an asset's initial value is $5,000 and it depreciates by $1,000 per year, the function is:

\[V( x) = - 1000x + 5000\]

Here, \(- 1000\) is the slope (annual depreciation) and \(5000\) is the initial value.

- Determining the Domain: The domain of the depreciation function is the period over which the depreciation occurs. If the asset depreciates over 7 years, the domain is:

\[[ 0, 7] \]

representing the time from purchase (year 0) to the end of the depreciation period (year 7).

Such knowledge enables the modeling and analysis of the depreciation of one's assets using linear functions, a broadly necessary skill for financiers and accountants. Turn to the UpStudy AI Homework Solver right away for help with this particular math concept or any other. With its great explanations and personalized support, UpStudy enables you to build a deep grip on every topic, helping to ensure success in the course. Just visit UpStudy now and redefine your learning experience!