Problem 74 - Algebra and Trigonometry

Question

The amount of a commodity sold is called the demand for the commodity. The demand \(D\) for a certain commodity is a function of the price given by $$D(p)=-3 p+150$$ (a) Find \(D^{-1} .\) What does \(D^{-1}\) represent? (b) Find \(D^{-1}(30 \text { ). What does your answer represent? }\)

Step-by-Step Solution

Answer

(a) \(D^{-1}(y) = \frac{150 - y}{3}\), it represents the price for a given demand. (b) \(D^{-1}(30) = 40\), the price for a demand of 30 is 40.

Step 1: Understand the Function

We are given the demand function for a commodity: \(D(p) = -3p + 150\). This function tells us the demand \(D\) based on the price \(p\). Our goal is to find the inverse of this function, \(D^{-1}\).

Step 2: Determine the Inverse Function

To find the inverse function \(D^{-1}(p)\), we need to solve the equation \(D(p) = y\) for \(p\) in terms of \(D\). Start with \(y = -3p + 150\) and solve for \(p\).

Step 3: Solve for p

Rearrange the equation: \(y = -3p + 150\) to find \(p\).Subtract 150 from both sides: \(y - 150 = -3p\).Divide both sides by -3: \(p = \frac{150 - y}{3}\).Thus, \(D^{-1}(y) = \frac{150 - y}{3}\).

Step 4: Interpret the Inverse Function

The inverse function \(D^{-1}(y)\) represents the price \(p\) that corresponds to a given demand \(y\). Essentially, it tells us what price is needed to achieve a specific demand.

Step 5: Calculate D^{-1}(30)

We need to find \(D^{-1}(30)\), which means determining the price when the demand is 30.Use the inverse function: \(D^{-1}(30) = \frac{150 - 30}{3}\).Calculate: \(D^{-1}(30) = \frac{120}{3} = 40\).

Step 6: Interpret the Result

The result \(D^{-1}(30) = 40\) signifies that when the demand for the commodity is 30, the price should be 40.

Key Concepts

Demand Function

A demand function represents the relationship between the price of a commodity and the quantity demanded by consumers. In simpler terms, it tells us how many units of a product people are willing to buy at different prices. The demand function given in our exercise is: \[ D(p) = -3p + 150 \] Here, \( D(p) \) is the demand and \( p \) is the price. This specific function shows that as the price increases, the demand decreases, which is a common scenario in real-world markets.

• \( -3p \): Indicates that for every unit increase in price, the demand decreases by 3 units.
• \( 150 \): Represents the maximum demand (i.e., the demand when the price is zero).

Understanding the demand function helps businesses and economists predict how changes in price might influence consumer purchasing behavior.

Price-Demand Relationship

The price-demand relationship is the direct connection between the price of a product and the quantity that consumers will purchase. This relationship is typically inverse, meaning as price goes up, demand goes down and vice versa. In the function \[ D(p) = -3p + 150 \] one can see this inverse relationship clearly represented by the negative coefficient of \( p \). This tells us that each price increase results in a proportionate decrease in demand.

• Consumers are usually more willing to buy a product at a lower price.
• The demand decreases as prices rise, which aligns with basic economic principles.

This relationship helps firms set prices strategically to maximize revenue or market share.

Function Inversion

Function inversion involves finding a new function that reverses the effect of the original function. For the demand function \( D(p) = -3p + 150 \), we want to determine the inverse function \( D^{-1}(y) \). Here’s how it’s done:- Start with the equation: \( y = -3p + 150 \).- Rearrange it to \( p = \frac{150 - y}{3} \). Now, \( D^{-1}(y) = \frac{150 - y}{3} \) is the inverse function, which means it calculates the price \( p \) given the demand \( y \). Function inversion is a valuable mathematical tool because it allows us to switch inputs and outputs, making it easier to solve different problems. Whether predicting price based on demand or vice versa, both functions give us essential economic insights.

Mathematical Interpretation

Mathematical interpretation involves understanding what the figures and equations represent in real-life scenarios. In this case, interpreting \( D^{-1}(y) \) informs us of the price needed to reach a specific demand level. For example:\[ D^{-1}(30) = \frac{150 - 30}{3} = 40 \] This numerical result tells us that a price of 40 will lead to a demand for 30 units.

• The inverse helps to not only predict necessary pricing strategies but also to understand consumer behavior and potential sales volume at different price points.
• It moves beyond static numbers and gives businesses the power to make data-driven decisions.

By converting mathematical expressions into actionable strategies, companies can optimize their pricing models to reflect consumer preferences and maximize profits more effectively.