Question

The number, \(N,\) of buyers looking to buy a home in a particular city is related to the average price, \(p,\) of a home in that city by the function \(N(p)=\frac{500000}{p} .\) Explain how the values of the function behave as the value of \(p\) changes and what the behaviour means in this situation.

Short answer

As the average home price increases, the number of buyers decreases and vice versa.

Step-by-step solution

- Understand the function

The given function is related to the number of buyers, N(p), and the average price of a home, p. It's in the form of a hyperbolic function: \[ N(p) = \frac{500000}{p} \]. The numerator, 500000, is a constant, which means N(p) changes as p changes according to this formula.

- Analyze the behavior for increasing values of p

As the average price, p, increases, the denominator of the function increases. Since the numerator remains constant, N(p) will decrease. In simpler terms, as homes become more expensive, fewer buyers are looking to buy homes.

- Analyze the behavior for decreasing values of p

As the average price, p, decreases, the denominator of the function decreases. Consequently, N(p) will increase. This means that as homes become cheaper, more buyers are looking to buy homes.

- Consider the limits of the function

When the price of a home, p, approaches zero, the number of buyers, N(p), theoretically approaches infinity, implying a very high demand which is unrealistic in reality but mathematically correct. Conversely, as the price increases without bound, N(p) approaches zero, indicating no buyers at extremely high prices.

- Interpret the real-world meaning

The function indicates an inverse relationship between the price of homes and the number of buyers. Practically, it means that more affordable homes attract more buyers, while pricier homes dissuade buyers.

Key concepts

Inverse Function

An inverse function essentially 'reverses' the effect of the original function. In simpler terms, if you have a function that takes an input and gives an output, the inverse function takes the output and returns to the original input. For the function given in the exercise, \(N(p) = \frac{500000}{p}\), it shows the relationship between the number of buyers, \(N\), and the average home price, \(p\).
Here, as the price \(p\) increases, the number of buyers \(N(p)\) decreases, and vice versa. This characteristic can be noted as an 'inverse relationship'.
To find an inverse function, you'd solve the original function for the input variable. So, for \(y = N(p) = \frac{500000}{p}\), you would swap \(y\) and \(p\) to get \(p = \frac{500000}{y}\). It indicates how the price of homes changes when the number of buyers is set.
Understanding how inverse functions work helps in many real-world applications, including economics and real estate.

Hyperbolic Function

A hyperbolic function is a type of mathematical function that has a significant property: it never touches a zero value and represents a reciprocal relationship. The given function in the exercise, \(N(p) = \frac{500000}{p}\), is a perfect example of a hyperbolic function.
This type of function is useful in modeling scenarios where one quantity varies inversely as another quantity. In real estate, it helps explain how demand for homes changes with price.
The general form of the hyperbolic function is \(f(x) = \frac{k}{x}\), where \(k\) is a constant. Here, our constant is 500000.
As observed:

• When \(p\) (price of homes) increases, \(N(p)\) (number of buyers) decreases.
• When \(p\) decreases, \(N(p)\) increases.

This pattern shows why hyperbolic functions are vital in understanding relationships like supply and demand.

Economic Demand

Economic demand refers to the willingness and ability of consumers to buy goods and services at different prices. The relationship between price and demand is a core concept in economics.
The given exercise directly ties to the law of demand, which states that as the price of a good rises, the quantity demanded generally falls, and as the price falls, the quantity demanded generally rises.
Our equation \(N(p) = \frac{500000}{p}\) clearly represents this law. Here's a breakdown:

• When home prices decrease, more buyers are willing and able to purchase homes \(N(p) increases\).
• Conversely, when home prices increase, fewer buyers are in the market \(N(p) decreases\).

This inverse relationship is fundamental for real estate markets and economic planning.
In essence, understanding economic demand helps sellers price homes strategically and helps policymakers in planning housing supply.