Problem 84
Question
Demand Function The amount of a commodity that is 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 \(D^{-1}(30) .\) What does your answer represent?
Step-by-Step Solution
Verified Answer
(a) \(D^{-1}(q) = \frac{150 - q}{3}\), the price as a function of demand.
(b) \(D^{-1}(30) = 40\), meaning 30 units are demanded at a price of $40.
1Step 1: Set up the function equation
The given demand function is \( D(p) = -3p + 150 \). To find the inverse, we'll start by setting \( D(p) = q \), where \( q \) represents the quantity demanded. Our equation becomes \( q = -3p + 150 \).
2Step 2: Solve for p in terms of q
To find \( D^{-1}(q) \), we need to express \( p \) in terms of \( q \). Begin by isolating \( p \) in the equation \( q = -3p + 150 \):Subtract 150 from both sides: \( q - 150 = -3p \).Now, divide both sides by -3 to solve for \( p \): \( p = \frac{150 - q}{3} \).
3Step 3: Write the inverse function
The equation \( p = \frac{150 - q}{3} \) represents the inverse function. Therefore, the inverse function \( D^{-1}(q) \), where \( q \) is the quantity demanded, is:\[ D^{-1}(q) = \frac{150 - q}{3} \]The inverse function \( D^{-1} \) represents the price \( p \) as a function of demand \( q \).
4Step 4: Compute D^{-1}(30)
Substitute \( q = 30 \) into the inverse function:\[ D^{-1}(30) = \frac{150 - 30}{3} \]Simplify the expression:\[ D^{-1}(30) = \frac{120}{3} = 40 \]So, \( D^{-1}(30) = 40 \).
5Step 5: Interpret the result of D^{-1}(30)
The calculated value \( D^{-1}(30) = 40 \) represents the price that corresponds to a quantity demanded of 30 units. In other words, when the quantity of the commodity demanded is 30 units, the price is $40.
Key Concepts
Inverse FunctionQuantity DemandedPrice-Quantity RelationshipSolving Equations
Inverse Function
An inverse function essentially reverses the roles of inputs and outputs in a given function. In simple terms, if you have a function that maps inputs to outputs, the inverse maps those outputs back to the original inputs. This concept is particularly handy when discussing demand functions in economics, as it allows for understanding how different prices impact consumer demand.
For a demand function like \( D(p) = -3p + 150 \), the goal when finding its inverse is to express the price, \( p \), as a function of the quantity demanded, \( q \). This means you are looking for \( D^{-1}(q) \), which tells you what price would generate each level of demand.
The process involves solving the equation for \( p \) in terms of \( q \). This flips the function, letting you substitute easily between price and demand quantities.
For a demand function like \( D(p) = -3p + 150 \), the goal when finding its inverse is to express the price, \( p \), as a function of the quantity demanded, \( q \). This means you are looking for \( D^{-1}(q) \), which tells you what price would generate each level of demand.
The process involves solving the equation for \( p \) in terms of \( q \). This flips the function, letting you substitute easily between price and demand quantities.
Quantity Demanded
The quantity demanded refers to the amount of a commodity that consumers wish to buy at a given price. In our initial demand function, this is represented by \( q \).
The demand equation \( D(p) = -3p + 150 \) indicates how quantity demanded is influenced by changes in price. It shows that demand decreases by 3 units for every 1 unit increase in price. The relationship here is negative, reflecting the general economic principle that higher prices tend to reduce consumer demand, while lower prices usually increase it.
Knowing how to compute the inverse function \( D^{-1} \) also helps explore this relationship in detail, by directly finding what price leads to a certain quantity being demanded.
The demand equation \( D(p) = -3p + 150 \) indicates how quantity demanded is influenced by changes in price. It shows that demand decreases by 3 units for every 1 unit increase in price. The relationship here is negative, reflecting the general economic principle that higher prices tend to reduce consumer demand, while lower prices usually increase it.
Knowing how to compute the inverse function \( D^{-1} \) also helps explore this relationship in detail, by directly finding what price leads to a certain quantity being demanded.
Price-Quantity Relationship
The price-quantity relationship is a fundamental concept captured in a demand function. It intricately links the price of a commodity to how much consumers are willing to buy at that price.
In our demand scenario, the function \( D(p) = -3p + 150 \) shows that as the price \( p \) rises, the quantity demanded \( q \) falls. Specifically, the coefficient \(-3\) indicates the rate at which demand decreases when price increases. Conversely, when price drops, demand tends to rise.
Switching to the inverse function, \( D^{-1}(q) = \frac{150 - q}{3} \), consumers and businesses alike can determine the price necessary to achieve a specific level of demand efficiently.
In our demand scenario, the function \( D(p) = -3p + 150 \) shows that as the price \( p \) rises, the quantity demanded \( q \) falls. Specifically, the coefficient \(-3\) indicates the rate at which demand decreases when price increases. Conversely, when price drops, demand tends to rise.
Switching to the inverse function, \( D^{-1}(q) = \frac{150 - q}{3} \), consumers and businesses alike can determine the price necessary to achieve a specific level of demand efficiently.
Solving Equations
Solving equations is a critical skill needed to manipulate functions, such as finding inverses in mathematical economics. This often involves simple algebraic manipulations to isolate the desired variable.
In the step-by-step solution provided, the purpose was to solve \( q = -3p + 150 \) for \( p \). This requires straightforward operations:
Finally, computing \( D^{-1}(30) \) by direct substitution, such as \( D^{-1}(30) = \frac{150 - 30}{3} = 40 \), helps solidify understanding of the outcome: At a demand level of 30 units, the commodity price is $40. This computation reinforces solving equations by substitution.
In the step-by-step solution provided, the purpose was to solve \( q = -3p + 150 \) for \( p \). This requires straightforward operations:
- Subtract 150 from both sides: \( q - 150 = -3p \).
- Divide by \(-3\): \( p = \frac{150 - q}{3} \).
Finally, computing \( D^{-1}(30) \) by direct substitution, such as \( D^{-1}(30) = \frac{150 - 30}{3} = 40 \), helps solidify understanding of the outcome: At a demand level of 30 units, the commodity price is $40. This computation reinforces solving equations by substitution.
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