Select The Correct Answer.Find The Inverse Of The Given Function.${ F(x) = -5x - 9 }$A. { F^{-1}(x) = 9x + 5 $}$B. { F^{-1}(x) = 5x - 9 $}$C. { F^{-1}(x) = \frac{x + 9}{-5} $} D . \[ D. \[ D . \[ F^{-1}(x) = \frac{x -

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. The inverse of a function essentially reverses the operation of the original function. In this article, we will focus on finding the inverse of a linear function, specifically the function $f(x) = -5x - 9$. We will explore the concept of inverse functions, the steps involved in finding the inverse of a linear function, and finally, we will select the correct answer from the given options.

Understanding Inverse Functions

An inverse function is a function that reverses the operation of the original function. In other words, if we have a function $f(x)$, its inverse function is denoted as $f^{-1}(x)$ and satisfies the property:

$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

This means that if we apply the original function to the inverse function, we get the original input, and vice versa.

Finding the Inverse of a Linear Function

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

1. Switch the variables: Switch the variables $x$ and $y$ in the original function.
2. Solve for $y$: Solve the resulting equation for $y$.
3. Interchange $x$ and $y$: Interchange $x$ and $y$ to obtain the inverse function.

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

Step 1: Switch the variables

Switch the variables $x$ and $y$ in the original function:

$y = -5x - 9$

Step 2: Solve for $y$

Solve the resulting equation for $y$:

$y + 9 = -5x$

$y = -5x - 9$

Step 3: Interchange $x$ and $y$

Interchange $x$ and $y$ to obtain the inverse function:

$x = -5y - 9$

$y = \frac{x + 9}{-5}$

Therefore, the inverse function of $f(x) = -5x - 9$ is:

$f^{-1}(x) = \frac{x + 9}{-5}$

Selecting the Correct Answer

Now that we have found the inverse function, let's compare it with the given options:

A. $f^{-1}(x) = 9x + 5$ B. $f^{-1}(x) = 5x - 9$ C. $f^{-1}(x) = \frac{x + 9}{-5}$ D. $f^{-1}(x) = \frac{x - 9}{-5}$

Based on our calculation, the correct answer is:

C. $f^{-1}(x) = \frac{x + 9}{-5}$

Conclusion

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Introduction

In our previous article, we discussed the concept of inverse functions and how to find the inverse of a linear function. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on inverse functions.

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

A: The purpose of finding the inverse of a function is to reverse the operation of the original function. This is useful in various applications, such as solving equations, graphing functions, and modeling real-world phenomena.

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 other words, if a function passes the horizontal line test, it has an inverse.

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

A: A function and its inverse are two different functions that are related to each other. The function takes an input value and produces an output value, while its inverse takes the output value and produces the original input value.

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

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

1. Switch the variables: Switch the variables $x$ and $y$ in the original function.
2. Solve for $y$: Solve the resulting equation for $y$.
3. Interchange $x$ and $y$: Interchange $x$ and $y$ to obtain the inverse function.

Q: Can a function have more than one inverse?

A: No, a function cannot have more than one inverse. The inverse of a function is unique and is denoted as $f^{-1}(x)$.

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

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

1. Graph the original function: Graph the original function on a coordinate plane.
2. Reflect the graph: Reflect the graph of the original function across the line $y = x$ to obtain the graph of the inverse function.

Q: What is the relationship between the domain and range of a function and its inverse?

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

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

A: No, a function cannot have an inverse if it is not one-to-one. A function must pass the horizontal line test to have an inverse.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts and provide additional information on inverse functions. We hope that this article has been helpful in understanding the concept of inverse functions and how to find the inverse of a function. If you have any further questions, please feel free to ask.