Problem 17 - Algebra Form and Function

Question

Let \(g(t)\) give the market value (in \(\$ 1000\) s) of a house in year \(t\). Say what the following statements tell you about the house. \(g(5)-g(0)=30\)

Step-by-Step Solution

Answer

Answer: The statement represents that the market value of the house increased by \(\$30,000\) between the years \(0\) and \(5\).

Step 1: Interpret the Function

The function \(g(t)\) represents the market value of a house in year \(t\). This means that \(g(t)\) gives the market value (in \(\$ 1000\) s) of a house in a given year.

Step 2: Identify the Years

In the given statement, we have \(g(5) - g(0) = 30\). Here, \(g(5)\) represents the market value of the house in year \(5\), and \(g(0)\) represents the market value of the house in year \(0\) (initial value).

Step 3: Interpret the Difference

The statement \(g(5) - g(0) = 30\) tells us the difference in market value of the house between year \(5\) and the initial year, year \(0\). This difference is equal to \(30\), which means that the market value has increased by \(30 \times \$1000 = \$30,000\) over these years.

Step 4: Summarize the Interpretation

The given statement \(g(5) - g(0) = 30\) tells us that the market value of the house increased by \(\$30{,}000\) between the years \(0\) and \(5\).

Key Concepts

Market Value

The market value is essentially the current worth of a property, such as a house, in a given market. In our function \(g(t)\), the market value is quantified in thousands of dollars and represented for different years by varying the variable \(t\). The function \(g(t)\) indicates the financial value of the house at specific times, helping us understand how the property's worth changes over time. Understanding market value can help potential buyers or sellers make informed decisions. For instance, if the market value increases, it might be a good time for a homeowner to sell the property for a profit. Conversely, a decrease might suggest waiting for better market conditions or improvements to increase the house's worth. In this context, when we discuss market value, we're examining how much someone would realistically pay for the house in a specified year, based on the function \(g(t)\).

Difference Interpretation

The concept of difference interpretation is crucial in analyzing how values change over a time period. In the function \(g(t)\), when we speak of \(g(5) - g(0) = 30\), we are looking at how much the value of the house has changed from the initial year (year 0) to year 5. Looking at this calculation more closely:

• \(g(5)\) is the house's market value at year 5.
• \(g(0)\) is the initial market value at year 0.
• The difference, or \(g(5) - g(0)\), tells us how much the value has increased or decreased over this time frame.

Here, the difference is 30, which translates to a \(\$30,000\) increase over those five years. This interpretation allows us to quantify the growth in the house's value and provides insights into its appreciation rate and the real estate market's health.

Yearly Change

Yearly change provides insights into how consistently, and at what rate, a property's market value is evolving each year. Interpreting yearly change in the context of \(g(t)\), where \(g(5) - g(0) = 30\), can be extremely valuable for forecasting future trends or making investment decisions.To understand the yearly change, we break down the total change over the given period:

• The total increase over 5 years is \(\\(30,000\).
• Dividing this increase by the number of years (5), we find the average yearly increase.
• This calculation: \(\frac{30,000}{5} = \\)6,000\) per year.

This yearly change suggests that on average, the market value of the house rose by \(\$6,000\) each year. It also hints at a steady increase, but keep in mind that actual yearly increases might fluctuate due to market conditions or property improvements. Regular analysis can help in understanding whether this trend will continue or if strategic decisions should be made to improve or capitalize on property investments.