Problem 105 - A Graphical Approach to College Algebra

Question

Suppose \(f(x)\) is the number of cars that can be built for \(x\) dollars. What does \(f^{-1}(1000)\) represent?

Step-by-Step Solution

Answer

\(f^{-1}(1000)\) represents the cost to produce 1000 cars.

Step 1: Understanding the function

The function \(f(x)\) represents the number of cars that can be constructed when \(x\) dollars are spent. Essentially, \(f(x)\) correlates the monetary investment to car production.

Step 2: Defining the inverse function

The inverse function, \(f^{-1}(y)\), reverses the roles of inputs and outputs compared to \(f(x)\). Specifically, \(f^{-1}(y)\) represents how many dollars are required to produce \(y\) cars.

Step 3: Interpreting the given problem

We are asked to interpret what \(f^{-1}(1000)\) represents. This means finding the value of \(x\) such that \(f(x) = 1000\), or in simpler terms, the dollars needed to manufacture 1000 cars.

Key Concepts

Function Notation

Understanding function notation is essential when working with functions and their inverses. In mathematics, a function like \( f(x) \) represents a relation where each input \( x \) is linked to a unique output \( f(x) \). The notation \( f(x) \) is used to describe the function in terms of its input and output

• The letter \( f \) is often used generically to denote a function, but it can be replaced by any letter.
• The expression within the parentheses, \( x \), indicates the variable or the input of the function.

This notation helps us understand the relationship between different variables involved in the function. It is not just about plugging in numbers; it maps a specific input to its respective output, which is a core concept in algebra and calculus.

Car Production Model

When discussing the car production model in this exercise, we refer to the function \( f(x) \). This model demonstrates how a specific amount of money \( x \) correlates to the manufacturing of cars. In such models:

• \( f(x) \) translates monetary investment into car numbers.
• It defines a clear relationship between spending and production output.

This model can be used to optimize production plans by predicting how investments affect the number of cars produced.
It provides insights into resource allocation by showing manufacturers the amount they need to invest to meet production goals.

Mathematical Interpretation

Mathematical interpretation allows us to translate real-world questions into mathematical expressions and solutions. In this context, the question about \( f^{-1}(1000) \) asks us to invert the usual function process. The inverse, \( f^{-1} \), lets us find the original input \( x \):

• \( f^{-1}(y) \) means finding how much input leads to a particular output \( y \).
• For \( f^{-1}(1000) \), identify the dollars required to produce 1000 cars.

This approach inverts the function, serving as a useful tool in various fields like economics and engineering. It helps solve for inputs when outputs are known, providing clarity in decision-making processes.