Question:For The Function F ( X ) = X − 2 5 9 F(x)=\frac{\sqrt[5]{x-2}}{9} F ( X ) = 9 5 X − 2 ​ ​ , Find F − 1 ( X F^{-1}(x F − 1 ( X ].Answer Options:A. F − 1 ( X ) = ( 9 X ) 5 + 2 F^{-1}(x) = (9x)^5 + 2 F − 1 ( X ) = ( 9 X ) 5 + 2 B. F − 1 ( X ) = 9 ( X + 2 ) 5 F^{-1}(x) = 9(x+2)^5 F − 1 ( X ) = 9 ( X + 2 ) 5 C. F − 1 ( X ) = ( 9 ( X + 2 ) ) 5 F^{-1}(x) = (9(x+2))^5 F − 1 ( X ) = ( 9 ( X + 2 ) ) 5 D. F − 1 ( X ) = 9 X 5 + 2 F^{-1}(x) = 9x^5 + 2 F − 1 ( X ) = 9 X 5 + 2

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 the original function. 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 $x$. In this article, we will explore how to find the inverse function of a given function, using the function $f(x)=\frac{\sqrt[5]{x-2}}{9}$ as an example.

Understanding the Given Function

The given function is $f(x)=\frac{\sqrt[5]{x-2}}{9}$. This function takes an input $x$ and outputs a value $y$ by raising $x-2$ to the power of $\frac{1}{5}$ and then dividing the result by $9$. To find the inverse function, we need to reverse this process.

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

The first step in finding the inverse function is to replace $f(x)$ with $y$. This gives us the equation $y = \frac{\sqrt[5]{x-2}}{9}$.

Step 2: Swap $x$ and $y$

The next step is to swap $x$ and $y$. This gives us the equation $x = \frac{\sqrt[5]{y-2}}{9}$.

Step 3: Solve for $y$

Now, we need to solve for $y$. To do this, we can start by isolating the term $\sqrt[5]{y-2}$.

Isolating the Term $\sqrt[5]{y-2}$

We can start by multiplying both sides of the equation by $9$ to get rid of the fraction. This gives us $9x = \sqrt[5]{y-2}$.

Raising Both Sides to the Power of $5$

Next, we can raise both sides of the equation to the power of $5$ to get rid of the fifth root. This gives us $(9x)^5 = y-2$.

Adding $2$ to Both Sides

Finally, we can add $2$ to both sides of the equation to solve for $y$. This gives us $y = (9x)^5 + 2$.

Conclusion

In conclusion, the inverse function of $f(x)=\frac{\sqrt[5]{x-2}}{9}$ is $f^{-1}(x) = (9x)^5 + 2$. This function takes an input $x$ and outputs a value $y$ by raising $9x$ to the power of $5$ and then adding $2$.

Answer Options

The answer options are:

A. $f^{-1}(x) = (9x)^5 + 2$ B. $f^{-1}(x) = 9(x+2)^5$ C. $f^{-1}(x) = (9(x+2))^5$ D. $f^{-1}(x) = 9x^5 + 2$

Discussion

The correct answer is option A, $f^{-1}(x) = (9x)^5 + 2$. This is because the inverse function of $f(x)=\frac{\sqrt[5]{x-2}}{9}$ is $f^{-1}(x) = (9x)^5 + 2$, as we derived earlier.

Example Use Case

Suppose we want to find the inverse function of $f(x)=\frac{\sqrt[5]{x-2}}{9}$ and we are given the input $x=3$. We can plug this value into the inverse function to get $f^{-1}(3) = (9(3))^5 + 2 = 243^5 + 2$.

Code Implementation

Here is an example code implementation in Python to calculate the inverse function:

import math
def inverse_function(x):
return (9*x)**5 + 2
x = 3
result = inverse_function(x)
print(result)

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

A: The inverse function of a given function is a function that undoes the action of the original function. 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 follow these steps:

1. Replace $f(x)$ with $y$.
2. Swap $x$ and $y$.
3. Solve for $y$.

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

A: A function and its inverse are two functions that are related to each other. The function $f(x)$ maps an input $x$ to an output $y$, while the inverse function $f^{-1}(x)$ maps the output $y$ back to the input $x$.

Q: Can a function have more than one inverse?

A: No, a function cannot have more than one inverse. The inverse function is unique and is denoted by $f^{-1}(x)$.

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

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

Q: What is the notation for the inverse function?

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

Q: Can I use the inverse function to solve equations?

A: Yes, you can use the inverse function to solve equations. For example, if you have an equation $f(x) = y$, you can use the inverse function $f^{-1}(x)$ to solve for $x$.

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

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

1. Find the inverse function of each component function.
2. Combine the inverse functions to get the inverse function of the composite function.

Q: Can I use the inverse function to find the domain and range of a function?

A: Yes, you can use the inverse function to find the domain and range of a function. The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.

Q: How do I graph the inverse function?

A: To graph the 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: Can I use the inverse function to solve optimization problems?

A: Yes, you can use the inverse function to solve optimization problems. For example, if you want to maximize or minimize a function, you can use the inverse function to find the maximum or minimum value.

Q: How do I use the inverse function to solve systems of equations?

A: To use the inverse function to solve systems of equations, you need to follow these steps:

1. Find the inverse function of each equation.
2. Combine the inverse functions to get the inverse function of the system of equations.
3. Solve for the variables using the inverse function.

Q: Can I use the inverse function to solve differential equations?

A: Yes, you can use the inverse function to solve differential equations. For example, if you have a differential equation $y' = f(x)$, you can use the inverse function $f^{-1}(x)$ to solve for $y$.

Q: How do I use the inverse function to solve partial differential equations?

A: To use the inverse function to solve partial differential equations, you need to follow these steps:

1. Find the inverse function of each partial differential equation.
2. Combine the inverse functions to get the inverse function of the system of partial differential equations.
3. Solve for the variables using the inverse function.

Q: Can I use the inverse function to solve integral equations?

A: Yes, you can use the inverse function to solve integral equations. For example, if you have an integral equation $\int_{a}^{b} f(x) dx = y$, you can use the inverse function $f^{-1}(x)$ to solve for $f(x)$.

Q: How do I use the inverse function to solve functional equations?

A: To use the inverse function to solve functional equations, you need to follow these steps:

1. Find the inverse function of each functional equation.
2. Combine the inverse functions to get the inverse function of the system of functional equations.
3. Solve for the variables using the inverse function.

Q: Can I use the inverse function to solve stochastic equations?

A: Yes, you can use the inverse function to solve stochastic equations. For example, if you have a stochastic equation $y' = f(x) + \epsilon$, where $\epsilon$ is a random variable, you can use the inverse function $f^{-1}(x)$ to solve for $y$.

Q: How do I use the inverse function to solve stochastic differential equations?

A: To use the inverse function to solve stochastic differential equations, you need to follow these steps:

1. Find the inverse function of each stochastic differential equation.
2. Combine the inverse functions to get the inverse function of the system of stochastic differential equations.
3. Solve for the variables using the inverse function.

Q: Can I use the inverse function to solve stochastic partial differential equations?

A: Yes, you can use the inverse function to solve stochastic partial differential equations. For example, if you have a stochastic partial differential equation $y' = f(x) + \epsilon$, where $\epsilon$ is a random variable, you can use the inverse function $f^{-1}(x)$ to solve for $y$.

Q: How do I use the inverse function to solve stochastic functional equations?

A: To use the inverse function to solve stochastic functional equations, you need to follow these steps:

1. Find the inverse function of each stochastic functional equation.
2. Combine the inverse functions to get the inverse function of the system of stochastic functional equations.
3. Solve for the variables using the inverse function.

Q: Can I use the inverse function to solve stochastic integral equations?

A: Yes, you can use the inverse function to solve stochastic integral equations. For example, if you have a stochastic integral equation $\int_{a}^{b} f(x) dx = y + \epsilon$, where $\epsilon$ is a random variable, you can use the inverse function $f^{-1}(x)$ to solve for $f(x)$.

Q: How do I use the inverse function to solve stochastic differential equations with delay?

A: To use the inverse function to solve stochastic differential equations with delay, you need to follow these steps:

1. Find the inverse function of each stochastic differential equation with delay.
2. Combine the inverse functions to get the inverse function of the system of stochastic differential equations with delay.
3. Solve for the variables using the inverse function.

Q: Can I use the inverse function to solve stochastic partial differential equations with delay?

A: Yes, you can use the inverse function to solve stochastic partial differential equations with delay. For example, if you have a stochastic partial differential equation with delay $y' = f(x) + \epsilon$, where $\epsilon$ is a random variable, you can use the inverse function $f^{-1}(x)$ to solve for $y$.

Q: How do I use the inverse function to solve stochastic functional equations with delay?

A: To use the inverse function to solve stochastic functional equations with delay, you need to follow these steps:

1. Find the inverse function of each stochastic functional equation with delay.
2. Combine the inverse functions to get the inverse function of the system of stochastic functional equations with delay.
3. Solve for the variables using the inverse function.

Q: Can I use the inverse function to solve stochastic integral equations with delay?

A: Yes, you can use the inverse function to solve stochastic integral equations with delay. For example, if you have a stochastic integral equation with delay $\int_{a}^{b} f(x) dx = y + \epsilon$, where $\epsilon$ is a random variable, you can use the inverse function $f^{-1}(x)$ to solve for $f(x)$.

Q: How do I use the inverse function to solve stochastic differential equations with random coefficients?

A: To use the inverse function to solve stochastic differential equations with random coefficients, you need to follow these steps:

1. Find the inverse function of each stochastic differential equation with random coefficients.
2. Combine the inverse functions to get the inverse function of the system of stochastic differential equations with random coefficients.
3. Solve for the variables using the inverse function.

Q: Can I use the inverse function to solve stochastic partial differential equations with random coefficients?

A: Yes, you can use the inverse function to solve stochastic partial differential equations with random coefficients. For example, if you have a stochastic partial differential equation with random coefficients $y' = f(x) + \epsilon$, where $\epsilon$ is a random variable, you can use the inverse function $f^{-1}(x)$ to solve for $y$.

**Q: How do I use the inverse function to solve