91Ó°ÊÓ

The concept of a demand function is crucial in economics. It represents how much of a product consumers are willing to buy at a given price. In our example, the demand function is given by:\[ D(p) = -3p + 150 \]Here, \( D(p) \) represents the quantity demanded, while \( p \) is the price of the commodity. This equation indicates that for every unit increase in price, the demand decreases by 3 units. If the price were zero, the demand would hit its maximum at 150 units.This function is linear, which simplifies understanding how demand changes with price fluctuations. Such functions are instrumental in economics as they help businesses and economists understand and predict buyer behavior under different pricing strategies.

The relationship between price and demand is fundamental in determining how quantity demanded changes with price fluctuations. In the provided demand function \( D(p) = -3p + 150 \), this relationship is clear.- **Negative Coefficient:** The negative coefficient of \( -3 \) in the function suggests the inverse nature of the relationship; as prices increase, demand decreases.- **Maximum Demand:** The constant 150 represents the theoretical maximum demand when the price is zero. This helps quantify how sensitive the demand is to price changes.Understanding this relationship enables businesses to make informed decisions about pricing to maximize revenue. Balancing a price point that optimizes demand without sacrificing too much in price per unit sold is vital.

Solving equations is a critical skill in finding inverse functions. Here we sought to find the inverse of the demand function, \( D(p) \), denoted as \( D^{-1}(p) \).1. **Set up the equation:** First, we set the equation equal to demand \( D \), such that \( D = -3p + 150 \).2. **Isolate \( p \):** To express \( p \) in terms of \( D \), follow these steps: - Subtract 150 from both sides: \( D - 150 = -3p \). - Divide by -3: \( p = \frac{150 - D}{3} \).The solution \( D^{-1}(D) = \frac{150 - D}{3} \) allows us to calculate the price given the demand is \( D \). Solving for \( D^{-1}(30) \) specifically means placing 30 in for \( D \) and solving for \( p \), leading to the conclusion \( p = 40 \). This indicates the price when demand is 30 units.