Problem 89
Question
Answer each of the following 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
Verified Answer
The amount of dollars needed to build 1000 cars.
1Step 1: Understand the Function
The function, denoted as \(f(x)\), represents the number of cars that can be built for \(x\) dollars.
2Step 2: Interpret the Inverse Function
The inverse function, denoted as \(f^{-1}(y)\), gives the amount of dollars needed to build \(y\) cars.
3Step 3: Apply the Given Value to the Inverse Function
Given \(f^{-1}(1000)\), it means we need to find the amount of dollars required to build 1000 cars.
Key Concepts
Function InterpretationCost AnalysisPrecalculus Problems
Function Interpretation
In this exercise, we deal with a function, which is a fundamental concept in mathematics usually denoted as \( f(x) \). Here, \( f(x) \) represents the number of cars that can be built for \( x \) dollars.
This means that for any given amount of money \( x \), the function will tell us how many cars can be produced with that budget.
The reverse of this function, called the inverse function, is equally important. This inverse function is represented as \( f^{-1}(y) \), and it works backward. For any given number of cars \( y \), it tells us how much money is needed to produce that number of cars. This swap of roles between inputs and outputs is essential in understanding inverse functions.
For example, if \( f(5000) = 10 \), it means that with 5000 dollars, 10 cars can be produced. Conversely, \( f^{-1}(10) = 5000\) means it costs 5000 dollars to make 10 cars.
This means that for any given amount of money \( x \), the function will tell us how many cars can be produced with that budget.
The reverse of this function, called the inverse function, is equally important. This inverse function is represented as \( f^{-1}(y) \), and it works backward. For any given number of cars \( y \), it tells us how much money is needed to produce that number of cars. This swap of roles between inputs and outputs is essential in understanding inverse functions.
For example, if \( f(5000) = 10 \), it means that with 5000 dollars, 10 cars can be produced. Conversely, \( f^{-1}(10) = 5000\) means it costs 5000 dollars to make 10 cars.
Cost Analysis
Cost analysis involves understanding how different levels of spending influence the outcome. In this case, we analyze how varying amounts of dollars alter the number of cars produced.
Given \( f(x) \) as the function for the number of cars produced for \( x \) dollars, \( f^{-1}(1000) \) is particularly significant for cost analysis.
What this inverse function tells us is the financial requirement for a specific production target.
When you see \( f^{-1}(1000) \), it is asking for the exact amount of dollars needed to produce 1000 cars.
If we were to practically apply this, knowing that \( f^{-1}(1000) = x \) helps manufacturers and planners to budget accordingly. They can determine how much money they need for their production goals.
This is extremely useful for forecasting and maintaining efficient resource allocation.
Given \( f(x) \) as the function for the number of cars produced for \( x \) dollars, \( f^{-1}(1000) \) is particularly significant for cost analysis.
What this inverse function tells us is the financial requirement for a specific production target.
When you see \( f^{-1}(1000) \), it is asking for the exact amount of dollars needed to produce 1000 cars.
If we were to practically apply this, knowing that \( f^{-1}(1000) = x \) helps manufacturers and planners to budget accordingly. They can determine how much money they need for their production goals.
This is extremely useful for forecasting and maintaining efficient resource allocation.
Precalculus Problems
Precalculus problems often involve working with different types of functions and their inverses.
These problems help build the foundation for more complex calculus topics.
In our example, interpreting functions and inverse functions is a classic precalculus problem.
Understanding how \( f(x) \) and \( f^{-1}(x) \) work prepares students for broader topics in mathematics.
Consider the function \( f(x) \) as a machine where you input dollars and output car quantity.
The inverse function \( f^{-1}(y) \) flips this process – you input the car quantity and output the cost.
Mastering these types of problems requires practice and a solid grasp of the relationships between variables.
It is helpful to take small steps:
These problems help build the foundation for more complex calculus topics.
In our example, interpreting functions and inverse functions is a classic precalculus problem.
Understanding how \( f(x) \) and \( f^{-1}(x) \) work prepares students for broader topics in mathematics.
Consider the function \( f(x) \) as a machine where you input dollars and output car quantity.
The inverse function \( f^{-1}(y) \) flips this process – you input the car quantity and output the cost.
Mastering these types of problems requires practice and a solid grasp of the relationships between variables.
It is helpful to take small steps:
- First, understand the original function \( f(x) \).
- Next, know what the inverse function \( f^{-1}(x) \) represents.
- Finally, practice with concrete values to see how these functions behave.
Other exercises in this chapter
Problem 89
Solve each equation. $$x^{-2 / 3}+x^{-1 / 3}-6=0$$
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Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real com
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Simplify each power of i. $$i^{27}$$
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Solve each rational inequality. Write each solution set in interval notation.4 $$\frac{(5-3 x)^{2}}{(2 x-5)^{3}}>0$$
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