Problem 66 - Calculus

Question

The cost of producing \(q\) articles is given by the function \(C=f(q)=100+2 q\). (a) Find a formula for the inverse function. (b) Explain in practical terms what the inverse function tells you.

Step-by-Step Solution

Answer

Inverse function: \(f^{-1}(C) = \frac{C - 100}{2}\). It tells how many articles can be made for a given cost.

Step 1: Set Up the Equation for Inverting

To find the inverse function of the cost function \(C = f(q) = 100 + 2q\), start by replacing \(C\) with \(y\), giving us \(y = 100 + 2q\). Our goal is to express \(q\) in terms of \(C\), that is, solve for \(q\).

Step 2: Isolate the Variable q

Subtract 100 from both sides of the equation to start isolating \(q\): \[ y - 100 = 2q \]

Step 3: Solve for q

Divide both sides by 2 to solve for \(q\): \[ q = \frac{y - 100}{2} \] Since we initially set \(y = C\), substitute \(y\) back with \(C\): \[ q = \frac{C - 100}{2} \] This yields the inverse function \( f^{-1}(C) = \frac{C - 100}{2} \).

Step 4: Interpret the Inverse Function

The inverse function \(f^{-1}(C) = \frac{C - 100}{2}\) tells us how many articles \(q\) can be produced at a given cost \(C\). Practically, it answers the question: 'If I know the total cost to produce the articles, how many articles can be produced with that cost?'

Key Concepts

Understanding the Cost Function

In any production scenario, the cost function is a critical tool used by businesses to determine the total cost of creating a certain number of products. In our exercise, the cost function is given by the formula \(C = f(q) = 100 + 2q\). This simple linear equation can be broken down as follows:

• The constant \(100\) represents the fixed costs, such as rent or salaries, that do not change regardless of production levels.
• The term \(2q\) indicates variable costs that fluctuate with the number of items produced. Here, \(2\) is the cost per article, and \(q\) is the number of articles.

Understanding this breakdown allows businesses to project how costs change with production levels, which is essential for budgeting and pricing decisions.

Methods for Solving Equations

Solving equations, especially when dealing with inverse functions, involves systematically converting one form into another. To find the inverse of a function, we aim to solve for one variable in terms of another. Let's examine the steps using our cost function as an example.
Initially, the cost function is \(y = 100 + 2q\). Our goal is to isolate \(q\). Here's how to proceed:

• Subtract the fixed costs from both sides: \(y - 100 = 2q\).
• Divide by the variable cost per article to solve for \(q\): \(q = \frac{y - 100}{2}\).

In practice, these steps help us "undo" the operations surrounding \(q\) so that we can express it solely in terms of \(y\) or \(C\), thus providing us the inverse function.

Function Interpretation in Real World Scenarios

Function interpretation allows us to translate mathematical expressions into practical insights. In this specific exercise, we deal with an inverse function \(f^{-1}(C) = \frac{C - 100}{2}\). Understanding this expression can have real-world implications.
This inverse function tells us the number of articles \(q\) that can be produced for a given cost \(C\). Here's how this can be valuable:

• If a company wants to know how many items they can produce given a budget, this function provides a direct answer.
• It helps in assessing the efficiency of production plans and determining whether the costs align with financial goals.

Using the inverse function provides a reverse calculation, allowing businesses to plan production according to available resources. Thus, it is a cornerstone for strategic decision-making in production management.