For The Function $f(x)=\frac X^5}{9}$, Find $f^{-1}(x)$.Answer A. $f^{-1 (x)=(9 X)^5$ B. $ F − 1 ( X ) = 9 X 5 F^{-1}(x)=9 X^5 F − 1 ( X ) = 9 X 5 [/tex] C. $f^{-1}(x)=\sqrt[5]{\left(\frac{x}{9}\right)}$ D.

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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 f^(-1)(x) maps the output y back to the original input x. In this article, we will focus on finding the inverse function of a given function f(x) = x^5/9.

Understanding the Given Function

The given function is f(x) = x^5/9. This is a rational function, where the numerator is a polynomial of degree 5 and the denominator is a constant. To find the inverse function, we need to start by understanding the behavior of this function. The function is increasing for all values of x, and it has a horizontal asymptote at y = 0.

Steps to Find the Inverse Function

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

  1. Switch the x and y variables: The first step in finding the inverse function is to switch the x and y variables. This means that we replace x with y and y with x in the original function.
  2. Solve for y: Once we have switched the x and y variables, we need to solve for y. This involves isolating y on one side of the equation.
  3. Write the inverse function: Once we have solved for y, we can write the inverse function by replacing y with x.

Finding the Inverse Function

Let's apply these steps to the given function f(x) = x^5/9.

Step 1: Switch the x and y variables

We start by switching the x and y variables. This gives us:

x = (y^5)/9

Step 2: Solve for y

To solve for y, we need to isolate y on one side of the equation. We can do this by multiplying both sides of the equation by 9:

9x = y^5

Next, we take the fifth root of both sides of the equation to get:

y = (9x)^(1/5)

Step 3: Write the inverse function

Now that we have solved for y, we can write the inverse function by replacing y with x:

f^(-1)(x) = (9x)^(1/5)

Simplifying the Inverse Function

We can simplify the inverse function by using the property of exponents that states (am)n = a^(mn). In this case, we have:

f^(-1)(x) = (9x)^(1/5) = 9^(1/5) * x^(1/5) = (9x)^(1/5)

However, we can simplify it further by using the property of exponents that states a^(1/n) = (n√a). In this case, we have:

f^(-1)(x) = (9x)^(1/5) = (5√9x) = (5√(9x))

Conclusion

In this article, we have found the inverse function of the given function f(x) = x^5/9. The inverse function is f^(-1)(x) = (9x)^(1/5). We have also simplified the inverse function using the properties of exponents.

Final Answer

The final answer is:

f^(-1)(x) = (9x)^(1/5)

This is option A.

Introduction

In our previous article, we discussed how to find the inverse function of a given function f(x) = x^5/9. We followed the steps to switch the x and y variables, solve for y, and write the inverse function. In this article, we will answer some common questions related to finding the inverse function.

Q&A

Q1: What is the purpose of finding the inverse function?

A1: The purpose of finding the inverse function is to understand the relationship between two functions. The inverse function undoes the action of the original function, and it can be used to solve equations and find the value of a variable.

Q2: How do I know if a function has an inverse function?

A2: A function has an inverse function if it is one-to-one, meaning that each value of x corresponds to a unique value of y. If a function is one-to-one, it is invertible.

Q3: What is the difference between a function and its inverse?

A3: A function and its inverse are two different 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 original input x.

Q4: How do I find the inverse function of a rational function?

A4: To find the inverse function of a rational function, you need to follow the steps we discussed earlier: switch the x and y variables, solve for y, and write the inverse function.

Q5: Can I use a calculator to find the inverse function?

A5: Yes, you can use a calculator to find the inverse function. However, it's always a good idea to understand the steps involved in finding the inverse function, so you can verify the answer and make sure it's correct.

Q6: What if the inverse function is not a simple expression?

A6: If the inverse function is not a simple expression, you may need to use algebraic manipulations or numerical methods to find it. In some cases, you may need to use a computer algebra system (CAS) or a graphing calculator to find the inverse function.

Q7: Can I find the inverse function of a function that is not invertible?

A7: No, you cannot find the inverse function of a function that is not invertible. If a function is not one-to-one, it does not have an inverse function.

Q8: How do I know if the inverse function is correct?

A8: To verify the inverse function, you can plug in the original function and the inverse function into the equation and check if they are equal. You can also use a graphing calculator or a computer algebra system to verify the inverse function.

Conclusion

In this article, we have answered some common questions related to finding the inverse function. We have discussed the purpose of finding the inverse function, how to know if a function has an inverse function, and how to find the inverse function of a rational function. We have also discussed some common pitfalls and how to verify the inverse function.

Final Tips

  • Always follow the steps to find the inverse function: switch the x and y variables, solve for y, and write the inverse function.
  • Use algebraic manipulations or numerical methods to find the inverse function if it's not a simple expression.
  • Verify the inverse function by plugging in the original function and the inverse function into the equation.
  • Use a graphing calculator or a computer algebra system to verify the inverse function if you're unsure.

Final Answer

The final answer is:

  • The purpose of finding the inverse function is to understand the relationship between two functions.
  • A function has an inverse function if it is one-to-one.
  • The inverse function undoes the action of the original function.
  • You can use a calculator to find the inverse function, but it's always a good idea to understand the steps involved.
  • You cannot find the inverse function of a function that is not invertible.
  • You can verify the inverse function by plugging in the original function and the inverse function into the equation.