Finding The Inverse Of F(x) = (10/9)x + 11

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In mathematics, the inverse of a function essentially reverses the operation performed by the original function. If a function $f$ takes an input $x$ and produces an output $y$, then its inverse, denoted as $f^{-1}$, takes $y$ as an input and returns $x$. In simpler terms, if $f(x) = y$, then $f^{-1}(y) = x$. Finding the inverse of a function is a fundamental concept in algebra and calculus, with applications spanning various fields, including cryptography, computer science, and engineering. Understanding how to determine the inverse of a function is crucial for solving equations, simplifying expressions, and analyzing mathematical relationships. This article delves into the process of finding the inverse of a function, providing a step-by-step guide with clear explanations and examples. We will explore the underlying principles, discuss the necessary conditions for a function to have an inverse, and illustrate the procedure with a practical example. By the end of this guide, you will have a solid understanding of how to find the inverse of a function and be able to apply this knowledge to solve a variety of mathematical problems. Mastering this skill will not only enhance your mathematical proficiency but also open doors to more advanced concepts and applications in various disciplines.

Understanding Inverse Functions

Before diving into the steps of finding an inverse, it's essential to understand the concept of inverse functions and their properties. A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The inverse function, if it exists, essentially undoes the operation of the original function. Not all functions have inverses. For a function to have an inverse, it must be one-to-one, also known as an injective function. A one-to-one function is a function where each element of the range corresponds to exactly one element of the domain. Graphically, a function is one-to-one if it passes the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once. This condition ensures that each output value corresponds to a unique input value, which is necessary for the inverse function to be well-defined. If a function is not one-to-one, it may be possible to restrict its domain to make it one-to-one, thereby allowing for the existence of an inverse function on the restricted domain. 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. This relationship highlights the symmetrical nature of inverse functions. Understanding these fundamental concepts is crucial for successfully finding and working with inverse functions. The process involves algebraic manipulation and a clear understanding of the relationship between a function and its inverse. In the following sections, we will outline the steps involved in finding the inverse of a function and provide a detailed example to illustrate the process. By grasping these concepts, you will be well-equipped to tackle more complex mathematical problems involving inverse functions.

Steps to Find the Inverse of a Function

Finding the inverse of a function involves a series of algebraic steps that systematically reverse the operations performed by the original function. The following steps provide a clear and concise guide to finding the inverse of a function:

  1. Replace $f(x)$ with $y$: This step simplifies the notation and makes the equation easier to manipulate. Instead of working with the function notation $f(x)$, we use the variable $y$ to represent the output of the function. This substitution helps in the algebraic manipulations that follow.
  2. Swap $x$ and $y$: This is the key step in finding the inverse. By interchanging the roles of $x$ and $y$, we are essentially reversing the input-output relationship of the function. This reflects the fundamental property of inverse functions, where the input of the original function becomes the output of the inverse function, and vice versa.
  3. Solve for $y$: After swapping $x$ and $y$, the goal is to isolate $y$ on one side of the equation. This involves using algebraic techniques such as addition, subtraction, multiplication, division, and factoring to get $y$ by itself. The resulting expression will represent the inverse function.
  4. Replace $y$ with $f^-1}(x)$ This final step expresses the inverse function using the standard notation. The notation $f^{-1(x)$ explicitly indicates that the function is the inverse of $f(x)$. This notation helps to clearly distinguish between the original function and its inverse.

These steps provide a systematic approach to finding the inverse of a function. By following these steps carefully, you can successfully determine the inverse function for a wide range of functions. It is important to note that not all functions have inverses, and the steps may need to be modified depending on the complexity of the function. In the next section, we will apply these steps to a specific example to illustrate the process in detail. This practical example will solidify your understanding of how to find the inverse of a function and provide a clear framework for solving similar problems.

Applying the Steps: A Detailed Example

Let's apply the steps outlined above to find the inverse of the function $f(x) = rac{10}{9}x + 11$. This example will provide a clear demonstration of the process and help you understand how to apply the steps in practice.

  1. Replace $f(x)$ with $y$: We start by replacing $f(x)$ with $y$ in the equation: $y = rac{10}{9}x + 11$. This substitution simplifies the notation and prepares the equation for the next steps.
  2. Swap $x$ and $y$: Next, we swap $x$ and $y$ to reverse the roles of input and output: $x = rac{10}{9}y + 11$. This step is crucial in finding the inverse function as it reflects the inverse relationship between the original function and its inverse.
  3. Solve for $y$: Now, we need to isolate $y$ on one side of the equation. To do this, we first subtract 11 from both sides:$x - 11 = rac10}{9}y$ Next, we multiply both sides by $ rac{9}{10}$ to solve for $y$$ rac{910}(x - 11) = y$ Distributing the $ rac{9}{10}$ gives$y = rac{910}x - rac{99}{10}$ 4. Replace $y$ with $f^{-1}(x)$ Finally, we replace $y$ with $f^{-1(x)$ to express the inverse function in standard notation:$f^-1}(x) = rac{9}{10}x - rac{99}{10}$ To further simplify, we can rewrite the equation as$f^{-1(x) = rac{9x - 99}{10}$ Thus, the inverse of the function $f(x) = rac{10}{9}x + 11$ is $f^{-1}(x) = rac{9x - 99}{10}$. This example demonstrates how to systematically apply the steps to find the inverse of a function. By following these steps carefully, you can solve a wide range of problems involving inverse functions. In the next section, we will discuss how to verify your answer to ensure that the inverse function is correct.

Verifying the Inverse Function

After finding the inverse of a function, it's crucial to verify your result to ensure that it is correct. The most common method for verifying the inverse function is to use the composition of functions. If $f^{-1}(x)$ is the inverse of $f(x)$, then the following two conditions must hold:

  1. f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}

  2. f^{-1}(f(x)) = x$ for all $x$ in the domain of $f

This means that if you compose the function with its inverse (in either order), the result should be the identity function, which simply returns the input value. Let's verify the inverse function we found in the previous example, $f(x) = rac{10}{9}x + 11$ and $f^{-1}(x) = rac{9x - 99}{10}$.

First, let's compute $f(f{-1}(x))$:$f(f-1}(x)) = f{ rac{9x - 99}{10}} = rac{10}{9}( rac{9x - 99}{10}) + 11$ Simplifying the expression$f(f^{-1(x)) = rac10}{9} imes rac{9x - 99}{10} + 11 = rac{9x - 99}{9} + 11 = x - 11 + 11 = x$ Now, let's compute $f^{-1}(f(x))$$f^{-1(f(x)) = f^-1}( rac{10}{9}x + 11) = rac{9( rac{10}{9}x + 11) - 99}{10}$ Simplifying the expression$f^{-1(f(x)) = rac{9( rac{10}{9}x + 11) - 99}{10} = rac{10x + 99 - 99}{10} = rac{10x}{10} = x$ Since both $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ hold true, we can confidently say that $f^{-1}(x) = rac{9x - 99}{10}$ is indeed the inverse of $f(x) = rac{10}{9}x + 11$. This verification step is essential to ensure the accuracy of your result. By composing the function with its inverse and checking if the result is the identity function, you can confirm that you have correctly found the inverse. This method is a reliable way to catch any errors in your calculations and ensure that you have a solid understanding of the concept of inverse functions.

Conclusion

In this comprehensive guide, we have explored the concept of inverse functions and provided a detailed, step-by-step method for finding the inverse of a function. We began by understanding the fundamental definition of an inverse function and the necessary condition for a function to have an inverse, which is that it must be one-to-one. We then outlined the four key steps for finding the inverse: replacing $f(x)$ with $y$, swapping $x$ and $y$, solving for $y$, and replacing $y$ with $f^{-1}(x)$. To illustrate the process, we worked through a detailed example, finding the inverse of the function $f(x) = rac{10}{9}x + 11$. The steps were applied systematically, demonstrating how to manipulate the equation to isolate $y$ and express the inverse function in standard notation. Furthermore, we emphasized the importance of verifying the inverse function to ensure accuracy. By composing the function with its inverse and checking if the result is the identity function, we can confirm that the inverse function is correct. This verification step is crucial for avoiding errors and building confidence in our solution. Understanding inverse functions is a fundamental concept in mathematics with wide-ranging applications. Mastering the ability to find and verify inverse functions enhances your mathematical skills and opens doors to more advanced topics in algebra, calculus, and other fields. Whether you are a student learning these concepts for the first time or a professional using them in your work, this guide provides a solid foundation for understanding and working with inverse functions. By following the steps and practicing with various examples, you can develop a strong understanding of inverse functions and their applications. The knowledge and skills gained will not only improve your mathematical proficiency but also empower you to tackle complex problems in various disciplines. The inverse of function problems often appear in various mathematical contexts, making a thorough understanding of the concept essential for mathematical proficiency.

Select the correct answer.

What is the inverse of function $f$?

f(x)=109x+11f(x)=\frac{10}{9} x+11

A. $f^{-1}(x)=\frac{9 x+11}{10}$ B. $f^{-1}(x)=\frac{10 x-110}{9}$ C. $f^{-1}(x)=\frac{9 x-99}{10}$ D. Discussion category : mathematics

Solution

To find the inverse of the function $f(x) = \frac{10}{9}x + 11$, we follow these steps:

  1. Replace $f(x)$ with $y$:$y = \frac{10}{9}x + 11$
  2. Swap $x$ and $y$:$x = \frac{10}{9}y + 11$
  3. Solve for $y$:Subtract 11 from both sides:

    x - 11 = \frac{10}{9}y$Multiply both sides by $\frac{9}{10}$:$\frac{9}{10}(x - 11) = y$Distribute and simplify:$y = \frac{9}{10}x - \frac{99}{10}

  4. Replace $y$ with $f{-1}(x)$:$f{-1}(x) = \frac{9x - 99}{10}$

Therefore, the correct answer is:

C. $f^{-1}(x) = \frac{9 x - 99}{10}$

This detailed solution demonstrates the step-by-step process of finding the inverse of a linear function, ensuring a clear understanding of each step involved. This problem reinforces the methodology discussed earlier and provides a practical application of the concepts learned.