Consider The Function F ( X ) = 7 X − 21 F(x) = \sqrt{7x - 21} F ( X ) = 7 X − 21 ​ .Place The Steps For Finding F − 1 ( X F^{-1}(x F − 1 ( X ] In The Correct Order:1. $y = \sqrt{7x - 21}$2. $x = \sqrt{7y - 21}$3. $x^2 = 7y - 21$4. $x^2 + 21 = 7y$5.

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Given a function $f(x)$, its inverse function $f^{-1}(x)$ is a function that undoes the action of the original function. In this article, we will explore the steps involved in finding the inverse function of a square root function, specifically the function $f(x) = \sqrt{7x - 21}$.

Step 1: Write the Original Function

The original function is given as $f(x) = \sqrt{7x - 21}$. To find the inverse function, we need to interchange the roles of $x$ and $y$. This means that we will replace $x$ with $y$ and $y$ with $x$.

Step 2: Interchange the Roles of x and y

We start by writing the original function with $y$ as the input variable: $y = \sqrt{7x - 21}$. Now, we interchange the roles of $x$ and $y$ to get $x = \sqrt{7y - 21}$.

Step 3: Square Both Sides

To eliminate the square root, we square both sides of the equation: $x^2 = (\sqrt{7y - 21})^2$. This simplifies to $x^2 = 7y - 21$.

Step 4: Add 21 to Both Sides

Next, we add 21 to both sides of the equation to isolate the term involving $y$: $x^2 + 21 = 7y$.

Step 5: Solve for y

Finally, we solve for $y$ by dividing both sides of the equation by 7: $y = \frac{x^2 + 21}{7}$.

Conclusion

In conclusion, the steps for finding the inverse function of the square root function $f(x) = \sqrt{7x - 21}$ are:

1. Write the original function with $y$ as the input variable: $y = \sqrt{7x - 21}$.
2. Interchange the roles of $x$ and $y$ to get $x = \sqrt{7y - 21}$.
3. Square both sides of the equation to eliminate the square root: $x^2 = 7y - 21$.
4. Add 21 to both sides of the equation to isolate the term involving $y$: $x^2 + 21 = 7y$.
5. Solve for $y$ by dividing both sides of the equation by 7: $y = \frac{x^2 + 21}{7}$.

Introduction

In our previous article, we explored the steps involved in finding the inverse function of a square root function, specifically the function $f(x) = \sqrt{7x - 21}$. In this article, we will address some common questions and concerns related to finding the inverse function of a square root function.

Q: What is the purpose of finding the inverse function of a square root function?

A: The purpose of finding the inverse function of a square root function is to understand the relationship between the original function and its inverse. This is useful in various applications, such as solving equations, graphing functions, and analyzing the behavior of functions.

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 value corresponds to exactly one input value. In the case of a square root function, we need to ensure that the expression inside the square root is non-negative.

Q: What if the expression inside the square root is negative?

A: If the expression inside the square root is negative, the function is not defined for that input value. In this case, we need to restrict the domain of the function to ensure that the expression inside the square root is non-negative.

Q: Can I use the same steps to find the inverse function of any square root function?

A: Yes, the steps we outlined in our previous article can be used to find the inverse function of any square root function. However, you need to be careful when squaring both sides of the equation, as this can introduce extraneous solutions.

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

A: To verify that you have found the correct inverse function, you can check the following:

• The inverse function should be a one-to-one function.
• The inverse function should satisfy the property $f(f^{-1}(x)) = x$.
• The inverse function should be a function that undoes the action of the original function.

Q: Can I use technology to find the inverse function of a square root function?

A: Yes, you can use technology, such as graphing calculators or computer algebra systems, to find the inverse function of a square root function. However, it's still important to understand the underlying steps and concepts involved in finding the inverse function.

Conclusion

In conclusion, finding the inverse function of a square root function requires careful attention to detail and a thorough understanding of the underlying concepts. By following the steps outlined in our previous article and addressing common questions and concerns, you can develop a deeper understanding of inverse functions and their applications.

Common Mistakes to Avoid

• Failing to interchange the roles of $x$ and $y$ in the original function.
• Squaring both sides of the equation without checking for extraneous solutions.
• Not verifying that the inverse function satisfies the property $f(f^{-1}(x)) = x$.

Additional Resources

• Khan Academy: Inverse Functions
• Mathway: Inverse Functions
• Wolfram Alpha: Inverse Functions

By following these resources and avoiding common mistakes, you can develop a strong understanding of inverse functions and their applications.