Problem 20 - Algebra Form and Function

Question

The value, \(V,\) of a car that is \(a\) years old is given by \(V=f(a)=18,000-3000 a\). Find and interpret: (a) The domain (b) The range

Step-by-Step Solution

Answer

Answer: The domain of the function is \([0, \infty)\), which represents all possible ages of the car for which the function is defined. The range of the function is \((-\infty, 18,000]\), which represents all possible values of the car based on its age, with the car's value continuously decreasing as it gets older, having no lower limit.

Step 1: Find the Domain

The domain of a function consists of all possible input values for which the function is defined. In this case, the input variable, \(a\), represents the age of the car. Since a car cannot be of a negative age, we can conclude that the domain consists of all non-negative numbers: Domain \(= [0, \infty)\)

Step 2: Find the Range

The range of a function consists of all possible output values for the given domain. Since the value of the car (\(V\)) decreases as its age (\(a\)) increases, we should find the minimum and maximum values of the function for our domain. First, let's find the value of the car when it's brand new, which is when \(a = 0\): \(V = f(0) = 18,000 - 3000(0) = 18,000\) Next, let's find the limiting value of the car as its age approaches infinity: \(V = \lim_{a\rightarrow\infty} (18,000 - 3000a) = -\infty\) So, the range of possible values of the car go from the starting value of 18,000 and decrease without bound as the age of the car increases: Range \(= (-\infty, 18,000]\)

Step 3: Interpret the Domain and Range

The domain, \([0, \infty)\), represents all possible ages of the car for which the function is defined. In other words, the function can calculate the value of a car for any age from the time it's brand new (\(a = 0\)) and onwards into the future. The range, \((-\infty, 18,000]\), represents all possible values of the car based on its age. The car has its highest value of 18,000 when it's brand new, and its value continuously decreases as it gets older, with no lower limit to its value.

Key Concepts

Functions

A function is a special relationship where each input has a single output. In mathematics, we often represent functions with letters like \( f \), and specify the input variable in parentheses, such as \( f(a) \). Here, the function \( f(a) = 18,000 - 3,000a \) describes the value of a car based on its age.

• The input, \( a \), represents the age of the car.
• The output, \( V \), gives the car's value at age \( a \).

The function subtracts \( 3,000 \) from \( 18,000 \) for every year, which models how a car's value typically decreases over time.
This type of mathematical relationship helps us to not only calculate the car's value quickly but also to predict future values under the same conditions.

Linear Equations

Linear equations are equations that graph as straight lines. The equation \( f(a) = 18,000 - 3,000a \) is an example of a linear equation.

• Linear equations can be expressed in the form \( y = mx + b \).
• Here, \( y \) represents the output, while \( x \) is the input variable.
• \( m \) is the slope and \( b \) is the y-intercept.

In our function, the slope \( m = -3,000 \) indicates that the car’s value decreases by \( 3,000 \) each year.
The y-intercept \( b = 18,000 \) is the value of the car when \( a = 0 \). This confirms that when the car is brand new, its value is \( 18,000 \).
Understanding these components allows you to interpret how the value changes over time, simply by observing the slope and intercept.

Interpreting Mathematical Models

Interpreting mathematical models involves understanding the real-world implications of an equation or function. For this car value function \( f(a) \), we interpret specifics like domain and range to gain insights.

• The domain \([0, \infty)\) represents all possible ages of the car, from zero onward.
• The range \((-\infty, 18,000]\) reflects the model predicting the car's value decreasing as age increases.

By examining the model, we see:- At \( a = 0 \), the model gives the starting value \( 18,000 \) for a brand new car. - As \( a \) increases, \( V \) continues to decrease, modeling depreciation. Interpretation guides us to make informed decisions, such as when considering the future value of an asset like a car.
It shows how mathematical models are essential tools for making predictions and understanding trends over time.