For The Function F ( X ) = X + 4 3 7 F(x)=\frac{\sqrt[3]{x+4}}{7} F ( X ) = 7 3 X + 4 ​ ​ , Find F − 1 ( X F^{-1}(x F − 1 ( X ].Answer:A. F − 1 ( X ) = ( 7 ( X − 4 ) ) 3 F^{-1}(x) = (7(x-4))^3 F − 1 ( X ) = ( 7 ( X − 4 ) ) 3 B. F − 1 ( X ) = ( 7 X ) 3 − 4 F^{-1}(x) = (7x)^3 - 4 F − 1 ( X ) = ( 7 X ) 3 − 4 C. F − 1 ( X ) = 7 ( X − 4 ) 3 F^{-1}(x) = 7(x-4)^3 F − 1 ( X ) = 7 ( X − 4 ) 3 D. F − 1 ( X ) = ( 7 X − 4 ) 3 F^{-1}(x) = (7x-4)^3 F − 1 ( X ) = ( 7 X − 4 ) 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 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 focus on finding the inverse function of a given function $f(x)=\frac{\sqrt[3]{x+4}}{7}$.

Understanding the Given Function

The given function is $f(x)=\frac{\sqrt[3]{x+4}}{7}$. This function takes an input $x$, adds $4$ to it, takes the cube root of the result, and then divides it by $7$. To find the inverse function, we need to reverse this process.

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

Let's start by replacing $f(x)$ with $y$ in the given function. This gives us:

$$y = \frac{\sqrt[3]{x+4}}{7}$$

Step 2: Swap $x$ and $y$

To find the inverse function, we need to swap $x$ and $y$. This gives us:

$$x = \frac{\sqrt[3]{y+4}}{7}$$

Step 3: Solve for $y$

Now, we need to solve for $y$. To do this, we can start by multiplying both sides of the equation by $7$:

$$7x = \sqrt[3]{y+4}$$

Next, we can cube both sides of the equation to get rid of the cube root:

$$(7x)^3 = y+4$$

Finally, we can subtract $4$ from both sides of the equation to solve for $y$:

$$(7x)^3 - 4 = y$$

Step 4: Write the Inverse Function

Now that we have solved for $y$, we can write the inverse function as:

$$f^{-1}(x) = (7x)^3 - 4$$

Conclusion

In this article, we have found the inverse function of a given function $f(x)=\frac{\sqrt[3]{x+4}}{7}$. We started by replacing $f(x)$ with $y$, swapping $x$ and $y$, solving for $y$, and finally writing the inverse function. The inverse function is $f^{-1}(x) = (7x)^3 - 4$.

Answer

The correct answer is:

• B. $f^{-1}(x) = (7x)^3 - 4$

Discussion

The concept of inverse functions is crucial in understanding the relationship between two functions. In this article, we have seen how to find the inverse function of a given function by replacing $f(x)$ with $y$, swapping $x$ and $y$, solving for $y$, and finally writing the inverse function. The inverse function is a function that undoes the action of the original function, and it is essential in many mathematical applications.

Example Use Cases

The concept of inverse functions has many practical applications in mathematics and other fields. Here are a few examples:

• Graphing Functions: Inverse functions are used to graph functions and their inverses. By graphing the inverse function, we can visualize the relationship between the original function and its inverse.
• Solving Equations: Inverse functions are used to solve equations that involve functions. By using the inverse function, we can isolate the variable and solve for its value.
• Modeling Real-World Situations: Inverse functions are used to model real-world situations that involve functions. By using the inverse function, we can analyze and predict the behavior of the system.

Conclusion

Q: What is an inverse function?

A: An inverse 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: Why is it important to find the inverse function?

A: Finding the inverse function is important because it helps us to understand the relationship between two functions. It also helps us to solve equations that involve functions and to model real-world situations that involve functions.

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 $y$ in the given function.
2. Swap $x$ and $y$.
3. Solve for $y$.
4. Write the inverse function.

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

A: The main difference between a function and its inverse is that the function maps an input $x$ to an output $y$, while the inverse function 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 it is one-to-one, meaning that each output $y$ corresponds to only one input $x$. If a function is one-to-one, then it has an inverse.

Q: What is the notation for the inverse function?

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

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

A: Yes, you can use the inverse function to solve equations that involve functions. By using the inverse function, you can isolate the variable and solve for its value.

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

A: Inverse functions have many real-world applications, including:

• Graphing functions and their inverses
• Solving equations that involve functions
• Modeling real-world situations that involve functions
• Calculating the area and volume of shapes
• Determining the maximum and minimum values of functions

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 $y=x$.
3. The resulting graph is the graph of the inverse 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. By using the inverse function, you can determine the values of $x$ and $y$ that are in the domain and range of the function.

Conclusion

In conclusion, inverse functions are an essential concept in mathematics that helps us to understand the relationship between two functions. By following the steps outlined in this article, you can find the inverse function of a given function and use it to solve equations, model real-world situations, and graph functions.