For The Function $f(x)=(x-10)^{\frac{1}{3}}$, Find $f^{-1}(x)$.A. $f {-1}(x)=(x-10) 3$B. $ F − 1 ( X ) = X 3 + 10 F^{-1}(x)=x^3+10 F − 1 ( X ) = X 3 + 10 [/tex]C. $f {-1}(x)=x {\frac{1}{3}}+10$D. $f {-1}(x)=(x+10) 3$

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 cubic root function, specifically the function $f(x)=(x-10)^{\frac{1}{3}}$.

Understanding the Given Function

The given function is $f(x)=(x-10)^{\frac{1}{3}}$. This function takes an input $x$ and raises it to the power of $\frac{1}{3}$, subtracting $10$ from the result. To understand this function better, let's consider an example. Suppose we want to find the value of $f(27)$. Plugging in $x=27$ into the function, we get:

$$f(27)=(27-10)^{\frac{1}{3}}=(17)^{\frac{1}{3}}=3$$

As we can see, the function $f(x)$ takes an input $x$ and returns a value that is the cubic root of $(x-10)$.

Finding the Inverse Function

To find the inverse function $f^{-1}(x)$, we need to swap the roles of $x$ and $y$ in the original function. In other words, we need to solve the equation $y=(x-10)^{\frac{1}{3}}$ for $x$. To do this, we can start by raising both sides of the equation to the power of $3$:

$$y^3=(x-10)$$

Next, we can add $10$ to both sides of the equation to get:

$$y^3+10=x$$

Now, we can swap the roles of $x$ and $y$ to get:

$$f^{-1}(x)=x^3+10$$

Verifying the Inverse Function

To verify that the inverse function $f^{-1}(x)=x^3+10$ is correct, we can check if it satisfies the property of inverse functions. Specifically, we need to check if $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$. Let's start by checking the first property:

$$f(f^{-1}(x))=f(x^3+10)=(x^3+10-10)^{\frac{1}{3}}=x^{\frac{1}{3}}$$

As we can see, $f(f^{-1}(x))$ does not equal $x$. This is because the function $f(x)$ takes an input $x$ and raises it to the power of $\frac{1}{3}$, while the inverse function $f^{-1}(x)$ takes an input $x$ and raises it to the power of $3$. To fix this, we need to modify the inverse function to get:

$$f^{-1}(x)=(x-10)^3$$

Now, let's check the second property:

$$f^{-1}(f(x))=f^{-1}((x-10)^{\frac{1}{3}})=((x-10)^{\frac{1}{3}}-10)^3=x-10$$

As we can see, $f^{-1}(f(x))$ does not equal $x$. This is because the function $f^{-1}(x)$ takes an input $x$ and raises it to the power of $3$, subtracting $10$ from the result, while the original function $f(x)$ takes an input $x$ and raises it to the power of $\frac{1}{3}$, subtracting $10$ from the result. To fix this, we need to modify the inverse function to get:

$$f^{-1}(x)=x^{\frac{1}{3}}+10$$

Now, let's check the second property:

$$f^{-1}(f(x))=f^{-1}((x-10)^{\frac{1}{3}})=(x-10)^{\frac{1}{3}}+10=x$$

As we can see, $f^{-1}(f(x))$ equals $x$, which verifies that the inverse function $f^{-1}(x)=x^{\frac{1}{3}}+10$ is correct.

Conclusion

Q: What is the inverse function of a cubic root function?

A: The inverse function of a cubic root function is a function that undoes the action of the cubic root function. In other words, if the cubic root function takes an input $x$ and returns a value that is the cubic root of $(x-10)$, then the inverse function takes an input $x$ and returns a value that is the cube of $(x-10)$.

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

A: To find the inverse function of a cubic root function, you need to swap the roles of $x$ and $y$ in the original function. In other words, you need to solve the equation $y=(x-10)^{\frac{1}{3}}$ for $x$. To do this, you can start by raising both sides of the equation to the power of $3$:

$$y^3=(x-10)$$

Next, you can add $10$ to both sides of the equation to get:

$$y^3+10=x$$

Now, you can swap the roles of $x$ and $y$ to get:

$$f^{-1}(x)=x^3+10$$

Q: What is the difference between the inverse function and the original function?

A: The inverse function and the original function are related but distinct. The original function takes an input $x$ and returns a value that is the cubic root of $(x-10)$, while the inverse function takes an input $x$ and returns a value that is the cube of $(x-10)$. In other words, the inverse function "reverses" the action of the original function.

Q: How do I verify that the inverse function is correct?

A: To verify that the inverse function is correct, you need to check if it satisfies the property of inverse functions. Specifically, you need to check if $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$. If both of these conditions are true, then the inverse function is correct.

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

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

• Swapping the roles of $x$ and $y$ incorrectly
• Failing to raise both sides of the equation to the correct power
• Failing to add or subtract the correct value from both sides of the equation
• Failing to verify that the inverse function satisfies the property of inverse functions

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

A: Yes, you can use the inverse function to solve equations involving the original function. For example, if you have an equation of the form $f(x)=y$, you can use the inverse function to solve for $x$. To do this, you can plug in the value of $y$ into the inverse function and solve for $x$.

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

A: The inverse function has many real-world applications, including:

• Physics: The inverse function is used to describe the motion of objects under the influence of a force.
• Engineering: The inverse function is used to design and optimize systems.
• Economics: The inverse function is used to model the behavior of economic systems.

Conclusion

In conclusion, the inverse function of a cubic root function is a function that undoes the action of the cubic root function. To find the inverse function, you need to swap the roles of $x$ and $y$ in the original function and solve for $x$. You can verify that the inverse function is correct by checking if it satisfies the property of inverse functions. The inverse function has many real-world applications, including physics, engineering, and economics.