Given The Coordinates For The Function Below, Which Of The Following Are Coordinates For Its Inverse?$\[ \begin{array}{|c|c|} \hline \text{Miles Traveled} & \text{Miles To Go} \\ \hline 0 & 650 \\ \hline 100 & 550 \\ \hline 200 & 450 \\ \hline 340

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Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two variables. Given a function, its inverse is a function that undoes the action of the original function. In this article, we will explore the concept of inverse functions and determine which coordinates belong to the inverse of a given function.

Understanding Inverse Functions

An inverse function is a function that reverses the operation of the original function. In other words, if we have a function f(x) that maps x to y, then its inverse function f^(-1)(x) maps y back to x. The inverse function is denoted by f^(-1) or f^{-1}.

The Given Function

The given function is represented by the following table:

Miles Traveled Miles to Go
0 650
100 550
200 450
340 310

This function represents the relationship between the miles traveled and the miles to go. We can see that as the miles traveled increase, the miles to go decrease.

Finding the Inverse Coordinates

To find the inverse coordinates, we need to swap the x and y values in the given table. The new table will represent the relationship between the miles to go and the miles traveled.

Miles to Go Miles Traveled
650 0
550 100
450 200
310 340

However, we need to check if the new table represents the inverse function of the original function. To do this, we need to verify if the new table satisfies the condition of an inverse function.

Verifying the Inverse Function

For a function to be an inverse function, it must satisfy the following condition:

f(f^(-1)(x)) = x

In other words, if we apply the original function to the inverse function, we should get the original input.

Let's apply the original function to the inverse function:

f(f^(-1)(x)) = f(650) = 0 f(f^(-1)(100)) = f(550) = 100 f(f^(-1)(200)) = f(450) = 200 f(f^(-1)(340)) = f(310) = 340

As we can see, the original function applied to the inverse function returns the original input. Therefore, the new table represents the inverse function of the original function.

Conclusion

In conclusion, the coordinates (650, 0), (550, 100), (450, 200), and (310, 340) belong to the inverse of the given function.

Key Takeaways

  • The inverse function is a function that reverses the operation of the original function.
  • To find the inverse coordinates, we need to swap the x and y values in the given table.
  • The inverse function must satisfy the condition f(f^(-1)(x)) = x.

Further Reading

For more information on inverse functions, we recommend the following resources:

  • Khan Academy: Inverse Functions
  • Math Is Fun: Inverse Functions
  • Wolfram MathWorld: Inverse Function

References

  • [1] Larson, R. (2015). Calculus. Cengage Learning.
  • [2] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.

Frequently Asked Questions

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of the original function. In other words, if we have a function f(x) that maps x to y, then its inverse function f^(-1)(x) maps y back to x.

Q: How do I find the inverse coordinates of a function?

A: To find the inverse coordinates, you need to swap the x and y values in the given table. The new table will represent the relationship between the input and output of the function.

Q: What is the condition for a function to be an inverse function?

A: For a function to be an inverse function, it must satisfy the condition f(f^(-1)(x)) = x. In other words, if we apply the original function to the inverse function, we should get the original input.

Q: How do I verify if a function is an inverse function?

A: To verify if a function is an inverse function, you need to check if the new table satisfies the condition of an inverse function. You can do this by applying the original function to the inverse function and checking if it returns the original input.

Q: What are some common mistakes to avoid when finding the inverse coordinates of a function?

A: Some common mistakes to avoid when finding the inverse coordinates of a function include:

  • Swapping the x and y values incorrectly
  • Not checking if the new table satisfies the condition of an inverse function
  • Not verifying if the function is one-to-one (injective)

Q: What are some real-world applications of inverse functions?

A: Inverse functions have many real-world applications, including:

  • Physics: Inverse functions are used to describe the relationship between variables such as distance, velocity, and time.
  • Engineering: Inverse functions are used to design and optimize systems such as electrical circuits and mechanical systems.
  • Economics: Inverse functions are used to model the relationship between variables such as price and quantity demanded.

Q: How do I graph an inverse function?

A: To graph an inverse function, you need to swap the x and y values of the original function and then reflect the resulting graph across the line y = x.

Q: What are some common types of inverse functions?

A: Some common types of inverse functions include:

  • Inverse linear functions
  • Inverse quadratic functions
  • Inverse trigonometric functions

Q: How do I find the inverse of a composite function?

A: To find the inverse of a composite function, you need to use the chain rule and the inverse function theorem.

Q: What are some common mistakes to avoid when finding the inverse of a composite function?

A: Some common mistakes to avoid when finding the inverse of a composite function include:

  • Not using the chain rule correctly
  • Not applying the inverse function theorem correctly
  • Not checking if the resulting function is one-to-one (injective)

Conclusion

In conclusion, inverse functions are a fundamental concept in mathematics that have many real-world applications. By understanding how to find the inverse coordinates of a function and how to verify if a function is an inverse function, you can apply this knowledge to a wide range of problems in physics, engineering, economics, and other fields.

Key Takeaways

  • Inverse functions are a fundamental concept in mathematics that have many real-world applications.
  • To find the inverse coordinates of a function, you need to swap the x and y values in the given table.
  • To verify if a function is an inverse function, you need to check if the new table satisfies the condition of an inverse function.
  • Inverse functions have many real-world applications, including physics, engineering, and economics.

Further Reading

For more information on inverse functions, we recommend the following resources:

  • Khan Academy: Inverse Functions
  • Math Is Fun: Inverse Functions
  • Wolfram MathWorld: Inverse Function

References

  • [1] Larson, R. (2015). Calculus. Cengage Learning.
  • [2] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.

Note: The references provided are for illustrative purposes only and are not intended to be a comprehensive list of resources on the topic.