For The Function F ( X ) = ( X 5 10 ) 1 7 F(x)=\left(\frac{x^5}{10}\right)^{\frac{1}{7}} F ( X ) = ( 10 X 5 ​ ) 7 1 ​ , Find F − 1 ( X F^{-1}(x F − 1 ( X ].Possible Answers:A. F − 1 ( X ) = ( 10 X ) 7 5 F^{-1}(x)=\sqrt[5]{(10 X)^7} F − 1 ( X ) = 5 ( 10 X ) 7 ​ B. F − 1 ( X ) = ( 10 X 4 ) 7 F^{-1}(x)=(10 \sqrt[4]{x})^7 F − 1 ( X ) = ( 10 4 X ​ ) 7 C. F − 1 ( X ) = ( 10 X 3 ) 7 F^{-1}(x)=(\sqrt[3]{10 X})^7 F − 1 ( X ) = ( 3 10 X ​ ) 7 D.

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Given a function $f(x)$, the inverse function $f^{-1}(x)$ is a function that undoes the action of $f(x)$. In other words, if $f(x)$ maps an input $x$ to an output $y$, then $f^{-1}(x)$ maps the output $y$ back to the input $x$. In this article, we will focus on finding the inverse function of a given function $f(x)=\left(\frac{x^5}{10}\right)^{\frac{1}{7}}$.

Understanding the Given Function

The given function is $f(x)=\left(\frac{x^5}{10}\right)^{\frac{1}{7}}$. To find the inverse function, we need to start by understanding the properties of this function. The function is a composite function, where the inner function is $\frac{x^5}{10}$ and the outer function is $x^{\frac{1}{7}}$. We can rewrite the function as $f(x)=\frac{x^{\frac{5}{7}}}{\sqrt[7]{10}}$.

Finding the Inverse Function

To find the inverse function, we need to swap the roles of $x$ and $y$ and then solve for $y$. Let's start by writing the function as $y=\left(\frac{x^5}{10}\right)^{\frac{1}{7}}$. Now, we can swap the roles of $x$ and $y$ to get $x=\left(\frac{y^5}{10}\right)^{\frac{1}{7}}$.

Solving for $y$

Now, we need to solve for $y$. We can start by isolating the term with $y$ by getting rid of the exponent $\frac{1}{7}$. We can do this by raising both sides of the equation to the power of $7$. This gives us $x^7=\left(\frac{y^5}{10}\right)$.

Simplifying the Equation

Now, we can simplify the equation by multiplying both sides by $10$ to get rid of the fraction. This gives us $10x^7=y^5$.

Finding the Inverse Function

Now, we can find the inverse function by taking the fifth root of both sides of the equation. This gives us $y=\sqrt[5]{10x^7}$.

Simplifying the Inverse Function

We can simplify the inverse function by rewriting it as $y=(10x^7)^{\frac{1}{5}}$. This gives us $y=(10x^{\frac{7}{5}})^{\frac{1}{5}}$.

Final Answer

The final answer is $f^{-1}(x)=\sqrt[5]{(10 x)^7}$.

Conclusion

In this article, we have found the inverse function of a given function $f(x)=\left(\frac{x^5}{10}\right)^{\frac{1}{7}}$. We started by understanding the properties of the given function and then swapped the roles of $x$ and $y$ to get the inverse function. We then solved for $y$ by isolating the term with $y$ and simplifying the equation. Finally, we found the inverse function by taking the fifth root of both sides of the equation. The final answer is $f^{-1}(x)=\sqrt[5]{(10 x)^7}$.

Possible Answers

A. $f^{-1}(x)=\sqrt[5]{(10 x)^7}$

B. $f^{-1}(x)=(10 \sqrt[4]{x})^7$

C. $f^{-1}(x)=(\sqrt[3]{10 x})^7$

D. $f^{-1}(x)=(10 x)^{\frac{1}{7}}$

Discussion

The correct answer is A. $f^{-1}(x)=\sqrt[5]{(10 x)^7}$. This is because we found the inverse function by taking the fifth root of both sides of the equation, which gives us $y=\sqrt[5]{10x^7}$. This is the same as the answer A.

Mathematical Proof

To prove that the answer A is correct, we can start by writing the function as $y=\left(\frac{x^5}{10}\right)^{\frac{1}{7}}$. Now, we can swap the roles of $x$ and $y$ to get $x=\left(\frac{y^5}{10}\right)^{\frac{1}{7}}$. We can then raise both sides of the equation to the power of $7$ to get $x^7=\left(\frac{y^5}{10}\right)$. We can then multiply both sides by $10$ to get rid of the fraction, which gives us $10x^7=y^5$. Finally, we can take the fifth root of both sides of the equation to get $y=\sqrt[5]{10x^7}$. This is the same as the answer A.

Conclusion

In conclusion, we have found the inverse function of a given function $f(x)=\left(\frac{x^5}{10}\right)^{\frac{1}{7}}$. We started by understanding the properties of the given function and then swapped the roles of $x$ and $y$ to get the inverse function. We then solved for $y$ by isolating the term with $y$ and simplifying the equation. Finally, we found the inverse function by taking the fifth root of both sides of the equation. The final answer is $f^{-1}(x)=\sqrt[5]{(10 x)^7}$.

Introduction

In our previous article, we found the inverse function of a given function $f(x)=\left(\frac{x^5}{10}\right)^{\frac{1}{7}}$. In this article, we will answer some common questions related to finding the inverse function of a given function.

Q: What is the inverse function of a given function?

A: The inverse function of a given function $f(x)$ is a function that undoes the action of $f(x)$. In other words, if $f(x)$ maps an input $x$ to an output $y$, then the inverse function $f^{-1}(x)$ maps the output $y$ back to the input $x$.

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

A: To find the inverse function of a given function, you need to start by understanding the properties of the given function. You can then swap the roles of $x$ and $y$ and solve for $y$ to get the inverse function.

Q: What are the steps to find the inverse function of a given function?

A: The steps to find the inverse function of a given function are:

1. Understand the properties of the given function.
2. Swap the roles of $x$ and $y$.
3. Solve for $y$ to get the inverse function.

Q: How do I know if I have found the correct inverse function?

A: To know if you have found the correct inverse function, you can check if the inverse function satisfies the following conditions:

• The inverse function is a function that maps an output $y$ back to the input $x$.
• The inverse function is a one-to-one function, meaning that each output $y$ corresponds to only one input $x$.

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

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

• Not understanding the properties of the given function.
• Not swapping the roles of $x$ and $y$ correctly.
• Not solving for $y$ correctly.

Q: Can I use a calculator to find the inverse function of a given function?

A: Yes, you can use a calculator to find the inverse function of a given function. However, it is always a good idea to check your work by hand to make sure that you have found the correct inverse function.

Q: How do I graph the inverse function of a given function?

A: To graph the inverse function of a given function, you can use a graphing calculator or a computer program. You can also use a table of values to graph the inverse function.

Q: What are some real-world applications of finding the inverse function of a given function?

A: Some real-world applications of finding the inverse function of a given function include:

• Modeling population growth and decline.
• Modeling the spread of diseases.
• Modeling the behavior of electrical circuits.

Conclusion

In conclusion, finding the inverse function of a given function is an important concept in mathematics. By understanding the properties of the given function and following the steps to find the inverse function, you can find the inverse function of a given function. Remember to check your work by hand and to use a calculator or computer program to graph the inverse function.

Final Answer

The final answer is $f^{-1}(x)=\sqrt[5]{(10 x)^7}$.

Discussion

The correct answer is A. $f^{-1}(x)=\sqrt[5]{(10 x)^7}$. This is because we found the inverse function by taking the fifth root of both sides of the equation, which gives us $y=\sqrt[5]{10x^7}$. This is the same as the answer A.

Mathematical Proof

To prove that the answer A is correct, we can start by writing the function as $y=\left(\frac{x^5}{10}\right)^{\frac{1}{7}}$. Now, we can swap the roles of $x$ and $y$ to get $x=\left(\frac{y^5}{10}\right)^{\frac{1}{7}}$. We can then raise both sides of the equation to the power of $7$ to get $x^7=\left(\frac{y^5}{10}\right)$. We can then multiply both sides by $10$ to get rid of the fraction, which gives us $10x^7=y^5$. Finally, we can take the fifth root of both sides of the equation to get $y=\sqrt[5]{10x^7}$. This is the same as the answer A.

Conclusion

In conclusion, we have found the inverse function of a given function $f(x)=\left(\frac{x^5}{10}\right)^{\frac{1}{7}}$. We started by understanding the properties of the given function and then swapped the roles of $x$ and $y$ to get the inverse function. We then solved for $y$ by isolating the term with $y$ and simplifying the equation. Finally, we found the inverse function by taking the fifth root of both sides of the equation. The final answer is $f^{-1}(x)=\sqrt[5]{(10 x)^7}$.