Select The Correct Answer.What Is The Range Of The Inverse Of The Given Function?${ F(x) = \sqrt{x-2} }$A. { [2, \infty)$}$B. { (-\infty, 2]$}$C. { [-2, \infty)$}$D. { (2, \infty]$}$

Introduction

In mathematics, functions are used to describe relationships between variables. The inverse of a function is a way to "undo" the original function, essentially reversing its operation. In this article, we will explore the concept of the inverse of a function and how to find its range.

What is the Inverse of a Function?

The inverse of a function is denoted as $f^{-1}(x)$ and is defined as a function that takes the output of the original function and returns the input. In other words, if $f(x)$ is the original function, then $f^{-1}(x)$ is the inverse function that "reverses" the operation of $f(x)$.

Finding the Inverse of a Function

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

1. Replace $f(x)$ with $y$: This is done to simplify the notation and make it easier to work with.
2. Switch $x$ and $y$: This is done to start the process of reversing the function.
3. Solve for $y$: This is done to isolate the inverse function.

Example: Finding the Inverse of $f(x) = \sqrt{x-2}$

Let's follow the steps to find the inverse of the given function $f(x) = \sqrt{x-2}$.

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

$f(x) = \sqrt{x-2}$ becomes $y = \sqrt{x-2}$.

Step 2: Switch $x$ and $y$

Switching $x$ and $y$ gives us $x = \sqrt{y-2}$.

Step 3: Solve for $y$

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

$x = \sqrt{y-2} \Rightarrow x^2 = y-2 \Rightarrow y = x^2 + 2$

Therefore, the inverse of the function $f(x) = \sqrt{x-2}$ is $f^{-1}(x) = x^2 + 2$.

Finding the Range of the Inverse Function

Now that we have found the inverse function, we need to find its range. The range of a function is the set of all possible output values.

The Range of the Inverse Function

To find the range of the inverse function, we need to consider the domain of the original function. The domain of the original function is the set of all possible input values.

In this case, the domain of the original function $f(x) = \sqrt{x-2}$ is $[2, \infty)$, since the square root of a negative number is not defined.

The Range of the Inverse Function

Since the domain of the original function is $[2, \infty)$, the range of the inverse function is also $[2, \infty)$.

Conclusion

In conclusion, the range of the inverse of the given function $f(x) = \sqrt{x-2}$ is $[2, \infty)$.

Answer

The correct answer is:

A. $[2, \infty)$

Discussion

This problem requires a good understanding of functions and their inverses. The student needs to be able to follow the steps to find the inverse of a function and then find its range.

Tips and Tricks

• Make sure to follow the steps to find the inverse of a function.
• Consider the domain of the original function when finding the range of the inverse function.
• Use the correct notation and terminology when working with functions and their inverses.

Practice Problems

1. Find the inverse of the function $f(x) = 2x-3$.
2. Find the range of the inverse function $f^{-1}(x) = \frac{x-1}{2}$.
3. Find the inverse of the function $f(x) = \frac{1}{x+2}$.

Solutions

1. The inverse of the function $f(x) = 2x-3$ is $f^{-1}(x) = \frac{x+3}{2}$.
2. The range of the inverse function $f^{-1}(x) = \frac{x-1}{2}$ is $(-\infty, \infty)$.
3. The inverse of the function $f(x) = \frac{1}{x+2}$ is $f^{-1}(x) = \frac{1}{x}-2$.

Conclusion

Q: What is the inverse of a function?

A: The inverse of a function is a way to "undo" the original function, essentially reversing its operation. It is denoted as $f^{-1}(x)$ and is defined as a function that takes the output of the original function and returns the input.

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

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

1. Replace $f(x)$ with $y$: This is done to simplify the notation and make it easier to work with.
2. Switch $x$ and $y$: This is done to start the process of reversing the function.
3. Solve for $y$: This is done to isolate the inverse function.

Q: What is the range of the inverse function?

A: The range of the inverse function is the set of all possible output values. To find the range of the inverse function, you need to consider the domain of the original function.

Q: How do I determine the domain of the original function?

A: The domain of the original function is the set of all possible input values. To determine the domain of the original function, you need to consider the values that make the function undefined or undefined.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values, while the range of a function is the set of all possible output values.

Q: Can a function have multiple inverses?

A: No, a function can only have one inverse. The inverse of a function is a unique function that "reverses" the operation of the original function.

Q: Can a function have an inverse if it is not one-to-one?

A: No, a function must be one-to-one in order to have an inverse. A one-to-one function is a function that passes the horizontal line test, meaning that no horizontal line intersects the graph of the function in more than one place.

Q: How do I know if a function is one-to-one?

A: To determine if a function is one-to-one, you can use the horizontal line test. If a horizontal line intersects the graph of the function in more than one place, then the function is not one-to-one.

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

A: The inverse of a function is significant because it allows us to "undo" the original function, essentially reversing its operation. This is useful in many areas of mathematics and science.

Q: Can the inverse of a function be used to solve equations?

A: Yes, the inverse of a function can be used to solve equations. By using the inverse of a function, we can "undo" the original function and solve for the input value.

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

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

1. Replace the original function with its inverse: This is done to "undo" the original function.
2. Solve for the input value: This is done to find the value of the input that satisfies the equation.

Q: What are some common applications of the inverse of a function?

A: The inverse of a function has many applications in mathematics and science, including:

• Solving equations: The inverse of a function can be used to solve equations by "undoing" the original function.
• Graphing functions: The inverse of a function can be used to graph functions by "reversing" the operation of the original function.
• Modeling real-world phenomena: The inverse of a function can be used to model real-world phenomena, such as population growth and decay.

Conclusion

In conclusion, the inverse of a function is a powerful tool that allows us to "undo" the original function, essentially reversing its operation. By understanding the concept of the inverse of a function, we can solve equations, graph functions, and model real-world phenomena.