Select The Correct Answer.What Is The Inverse Of This Function?$f(x)=8 \sqrt{x}, \text{ For } X \geq 0$A. $f^{-1}(x)=\frac{1}{64} X^2, \text{ For } X \geq 0$ B. $f^{-1}(x)=8 X^2, \text{ For } X \geq 0$ C.

Introduction

In mathematics, an inverse function is a function that reverses the operation of another function. It is denoted by the symbol $f^{-1}(x)$ and is used to find the input value that produces a given output value. In this article, we will discuss the concept of inverse functions and learn how to find the inverse of a given function.

What is an Inverse Function?

An inverse function is a function that undoes the operation of another function. It is a one-to-one function that takes the output of the original function and returns the input value that produced that output. In other words, if $f(x)$ is a function, then its inverse function $f^{-1}(x)$ is a function that satisfies the following property:

$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

Finding the Inverse of a Function

To find the inverse of a function, we need to follow these steps:

1. Replace $f(x)$ with $y$ in the original function.
2. Swap the roles of $x$ and $y$.
3. Solve for $y$ in terms of $x$.

Finding the Inverse of the Given Function

The given function is $f(x) = 8 \sqrt{x}, \text{ for } x \geq 0$. To find the inverse of this function, we will follow the steps mentioned above.

Step 1: Replace $f(x)$ with $y$

$f(x) = 8 \sqrt{x}$

Replace $f(x)$ with $y$:

$y = 8 \sqrt{x}$

Step 2: Swap the roles of $x$ and $y$

Swap the roles of $x$ and $y$:

$x = 8 \sqrt{y}$

Step 3: Solve for $y$ in terms of $x$

Solve for $y$ in terms of $x$:

$x = 8 \sqrt{y}$

Square both sides:

$x^2 = 64y$

Divide both sides by 64:

$y = \frac{x^2}{64}$

Therefore, the inverse of the given function is $f^{-1}(x) = \frac{x^2}{64}, \text{ for } x \geq 0$.

Conclusion

In this article, we discussed the concept of inverse functions and learned how to find the inverse of a given function. We used the given function $f(x) = 8 \sqrt{x}, \text{ for } x \geq 0$ to illustrate the process of finding the inverse of a function. We replaced $f(x)$ with $y$, swapped the roles of $x$ and $y$, and solved for $y$ in terms of $x$ to find the inverse of the given function.

Answer

The correct answer is:

A. $f^{-1}(x)=\frac{1}{64} x^2, \text{ for } x \geq 0$

Discussion

The concept of inverse functions is an important topic in mathematics, and it has many applications in various fields such as physics, engineering, and economics. Inverse functions are used to solve equations, model real-world phenomena, and make predictions about future events.

In this article, we discussed the concept of inverse functions and learned how to find the inverse of a given function. We used the given function $f(x) = 8 \sqrt{x}, \text{ for } x \geq 0$ to illustrate the process of finding the inverse of a function.

If you have any questions or need further clarification on the concept of inverse functions, please feel free to ask.

References

• [1] "Inverse Functions" by Math Open Reference
• [2] "Inverse Functions" by Khan Academy
• [3] "Inverse Functions" by Wolfram MathWorld

Related Topics

• [1] "Functions" by Math Open Reference
• [2] "One-to-One Functions" by Khan Academy
• [3] "Inverse Functions" by Wolfram MathWorld

FAQs

• Q: What is an inverse function?
• A: An inverse function is a function that reverses the operation of another function.
• Q: How do I find the inverse of a function?
• A: To find the inverse of a function, you need to replace $f(x)$ with $y$, swap the roles of $x$ and $y$, and solve for $y$ in terms of $x$.
• Q: What is the inverse of the given function $f(x) = 8 \sqrt{x}, \text{ for } x \geq 0$?
• A: The inverse of the given function is $f^{-1}(x) = \frac{x^2}{64}, \text{ for } x \geq 0$.
  Inverse Functions: A Comprehensive Q&A Guide =====================================================

Introduction

Inverse functions are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and economics. In this article, we will provide a comprehensive Q&A guide on inverse functions, covering topics such as the definition of inverse functions, how to find the inverse of a function, and examples of inverse functions.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. It is denoted by the symbol $f^{-1}(x)$ and is used to find the input value that produces a given output value.

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

A: To find the inverse of a function, you need to follow these steps:

1. Replace $f(x)$ with $y$ in the original function.
2. Swap the roles of $x$ and $y$.
3. Solve for $y$ in terms of $x$.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two different functions that are related to each other. The function $f(x)$ and its inverse $f^{-1}(x)$ are two different functions that are used to find the input value that produces a given output value.

Q: How do I know if a function has an inverse?

A: A function has an inverse if it is one-to-one, meaning that each output value corresponds to only one input value. If a function is one-to-one, then it has an inverse.

Q: What is the notation for an inverse function?

A: The notation for an inverse function is $f^{-1}(x)$, where $f(x)$ is the original function.

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

A: To find the inverse of a composite function, you need to follow these steps:

1. Replace the composite function with a single function.
2. Find the inverse of the single function.
3. Replace the single function with the composite function.

Q: What is the relationship between a function and its inverse?

A: The relationship between a function and its inverse is that they are two different functions that are used to find the input value that produces a given output value. The function $f(x)$ and its inverse $f^{-1}(x)$ are two different functions that are used to find the input value that produces a given output value.

Q: How do I use inverse functions in real-world applications?

A: Inverse functions are used in various real-world applications such as physics, engineering, and economics. They are used to model real-world phenomena, solve equations, and make predictions about future events.

Q: What are some examples of inverse functions?

A: Some examples of inverse functions include:

• $f(x) = 2x$ and $f^{-1}(x) = \frac{x}{2}$
• $f(x) = x^2$ and $f^{-1}(x) = \sqrt{x}$
• $f(x) = \sin x$ and $f^{-1}(x) = \arcsin x$

Q: How do I graph an inverse function?

A: To graph an inverse function, you need to follow these steps:

1. Graph the original function.
2. Reflect the graph of the original function across the line $y = x$.

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

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

• Not following the steps to find the inverse of a function.
• Not checking if the function is one-to-one.
• Not using the correct notation for the inverse function.

Conclusion

In this article, we provided a comprehensive Q&A guide on inverse functions, covering topics such as the definition of inverse functions, how to find the inverse of a function, and examples of inverse functions. We hope that this guide has been helpful in understanding the concept of inverse functions and how to apply it in real-world applications.

References

• [1] "Inverse Functions" by Math Open Reference
• [2] "Inverse Functions" by Khan Academy
• [3] "Inverse Functions" by Wolfram MathWorld

Related Topics

• [1] "Functions" by Math Open Reference
• [2] "One-to-One Functions" by Khan Academy
• [3] "Inverse Functions" by Wolfram MathWorld

FAQs

• Q: What is an inverse function?
• A: An inverse function is a function that reverses the operation of another function.
• Q: How do I find the inverse of a function?
• A: To find the inverse of a function, you need to replace $f(x)$ with $y$, swap the roles of $x$ and $y$, and solve for $y$ in terms of $x$.
• Q: What is the notation for an inverse function?
• A: The notation for an inverse function is $f^{-1}(x)$, where $f(x)$ is the original function.