Problem 73 - Precalculus : Building Concepts and Connections

Question

Applications In this set of exercises, you will use inverse functions to study real-world problems. In economics, the demand function gives the price \(p\) as a function of the quantity \(q .\) One example of a demand function is \(p=100-0.1 q .\) However, mathematicians tend to think of the price as the input variable and the quantity as the output variable. How can you take this example of a demand function and express \(q\) as a function of p?

Step-by-Step Solution

Answer

The quantity q as a function of price p is \(q = 10*(100 - p)\).

Step 1: Understand the given function

The given function is \(p = 100 - 0.1q\). Here, \(p\) is the price and \(q\) is the quantity. However, we need to express \(q\) in terms of \(p\), essentially, we need to isolate \(q\) in the function.

Step 2: Manipulate the given function to isolate q

To do this, one can manipulate the given function such that quantity \(q\) is on one side and price \(p\) on the other. Start by moving the terms around: \(0.1q = 100 - p\)

Step 3: Solve for q

To isolate \(q\), we need to divide the expression \(100 - p\) on both sides by 0.1. This gives us the function \(q = \frac{{100 - p}}{{0.1}}.\)

Step 4: Simplify

The decimal in the function can be removed by multiplying the numerator and denominator by 10. This gives a final demand function of \(q = 10*(100 - p)\), which is easier to interpret.

Key Concepts

Demand Function

A demand function is a mathematical representation of how the quantity demanded of a good responds to changes in price. It is usually expressed as a function where price \( p \) is dependent on the quantity \( q \). In simple terms, it tells us how much of a product people are willing to buy at different prices.

• The demand function is typically written as \( p = f(q) \), indicating that price is the dependent variable and quantity is the independent variable.
• For example, in our exercise, the demand function is given as \( p = 100 - 0.1q \).

In economics, understanding demand functions is critical because they help businesses determine the optimal price to charge for their products. They also provide insights into how market demand might change if prices are adjusted.

To convert the given demand function into an inverse function, one must express quantity as a function of price. This involves a few algebraic manipulations to "reverse" the roles of \( p \) and \( q \). Here, by manipulating the equation \( p = 100 - 0.1q \) into \( q = \frac{100 - p}{0.1} \), we explicitly determine how quantity changes with price, which is essential for understanding consumer behavior.

Economic Modeling

Economic modeling is a process used to represent economic processes through mathematical equations or simulations. It allows economists to analyze and predict economic behaviors by setting up models that simulate real-world economic scenarios.

• Models often use functions like demand functions to depict relationships between different economic variables, such as price and quantity.
• These functions help model consumer demand, supply, pricing strategies, and overall economic equilibrium.

Using models, economists can experiment with different variables and assess potential outcomes without altering real-world economic conditions—which can be costly or impractical. In the context of the demand function presented, it allows an economist to predict how changing prices might affect quantities demanded without changing the actual market conditions.

Inverse functions in these models are especially useful. They help determine outcomes when typical roles of variables are reversed—such as solving for quantity given a price instead of vice versa. This flexibility makes economic modeling a powerful tool for policy-making, business strategy development, and economic forecasting.

Function Manipulation

Function manipulation involves altering the form of a function to achieve a desired configuration or to extract specific information. This is a foundational skill in both mathematics and economics, helping to discover different expressions of variable relationships.

• The manipulation often includes isolating a variable, expanding or factoring expressions, and simplifying formulations.
• In our case, the exercise detailed how to convert \( p = 100 - 0.1q \) into \( q = \frac{100 - p}{0.1} \), transitioning from a direct demand function to its inverse.

Such manipulations are critical when exploring inverse functions, as they allow economists and mathematicians to explore alternative perspectives and solutions within a given context. It's a skill that involves algebraic creativity and logical reasoning.

Getting comfortable with these transformations can demystify complex economic models and allow one to perform meaningful analyses. By effectively manipulating functions, one can derive insights on consumer behavior, market dynamics, and economic stability.