Finding The Inverse Function For F(x) = 5 - 9x

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In mathematics, determining the inverse of a function is a fundamental concept. The inverse function, denoted as f⁻¹(x), essentially undoes what the original function f(x) does. This article will delve into the process of finding the inverse function for the given relation f(x) = 5 - 9x, providing a step-by-step explanation and clarifying the underlying principles. Understanding inverse functions is crucial for various mathematical applications, including solving equations, analyzing graphs, and exploring more advanced topics in calculus and linear algebra. We will not only derive the correct inverse function but also discuss the implications of this result and how it relates to the original function. So, let's embark on this mathematical journey to demystify the concept of inverse functions.

Understanding Inverse Functions

Before we dive into the specific problem of finding the inverse of f(x) = 5 - 9x, let's solidify our understanding of what an inverse function truly represents. In simple terms, if a function f(x) takes an input x and produces an output y, then the inverse function f⁻¹(x) takes that output y and returns the original input x. This "undoing" action is the essence of an inverse function. Mathematically, this relationship can be expressed as f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. These equations highlight the core property of inverse functions: they reverse the effect of the original function.

To visualize this, consider a function as a machine that transforms an input into an output. The inverse function is another machine that reverses this transformation, taking the output back to its original input. Not all functions have inverses. A function must be one-to-one (also known as injective) to possess an inverse. A one-to-one function means that each input maps to a unique output, and conversely, each output corresponds to a unique input. This can be visually checked using the horizontal line test: if any horizontal line intersects the graph of the function at most once, then the function is one-to-one and has an inverse. The concept of inverse functions is not just a theoretical exercise; it has practical applications in cryptography, data analysis, and many other fields. For instance, in cryptography, inverse functions are used to encrypt and decrypt messages, ensuring secure communication. Understanding the properties and applications of inverse functions is therefore a valuable asset in various domains.

Steps to Find the Inverse Function

The process of finding the inverse function involves a few key steps. Let's outline these steps in a clear and concise manner before applying them to our specific function, f(x) = 5 - 9x. These steps provide a systematic approach to solving this type of problem, ensuring accuracy and understanding.

  1. Replace f(x) with y: This substitution simplifies the notation and makes the algebraic manipulations easier to follow. We essentially rewrite the function as y = 5 - 9x, which represents the same relationship but in a more convenient form for finding the inverse.
  2. Swap x and y: This is the crucial step where we reverse the roles of the input and output. By interchanging x and y, we are reflecting the function across the line y = x, which is the graphical representation of finding the inverse. After this step, our equation becomes x = 5 - 9y.
  3. Solve for y: Now, we need to isolate y on one side of the equation. This involves algebraic manipulation to get y in terms of x. The goal is to express y as a function of x, which will be our inverse function. In the equation x = 5 - 9y, we will add 9y to both sides and subtract x from both sides, then divide by 9.
  4. Replace y with f⁻¹(x): Finally, we replace y with the notation f⁻¹(x) to denote the inverse function. This gives us the final expression for the inverse function, which we can use to find the input value corresponding to a given output value of the original function. By following these steps meticulously, we can accurately determine the inverse of any invertible function. Understanding the logic behind each step is as important as performing the algebraic manipulations, as it helps in applying this method to different types of functions.

Applying the Steps to f(x) = 5 - 9x

Now, let's put our outlined steps into action and find the inverse function for f(x) = 5 - 9x. This is where the theoretical understanding transforms into a practical application. We will go through each step methodically, showing the algebraic manipulations involved and explaining the reasoning behind them. This will provide a clear and concise solution to the problem.

  1. Replace f(x) with y: Starting with f(x) = 5 - 9x, we substitute y for f(x), resulting in the equation y = 5 - 9x. This simple substitution sets the stage for the subsequent steps.
  2. Swap x and y: Next, we interchange x and y, which gives us x = 5 - 9y. This is the key step in finding the inverse, as it reverses the roles of input and output.
  3. Solve for y: Now, we need to isolate y in the equation x = 5 - 9y. First, subtract 5 from both sides: x - 5 = -9y. Then, divide both sides by -9: y = (x - 5) / -9. Simplifying this expression, we get y = (-x + 5) / 9, which can be further written as y = (-x/9) + (5/9). This is our expression for y in terms of x.
  4. Replace y with f⁻¹(x): Finally, we replace y with f⁻¹(x) to denote the inverse function. Therefore, the inverse function is f⁻¹(x) = (-x/9) + (5/9). This is the final answer, representing the inverse of the original function f(x) = 5 - 9x. By following these steps carefully, we have successfully found the inverse function.

The Solution and Its Implications

Having worked through the steps, we have found the inverse function for f(x) = 5 - 9x to be f⁻¹(x) = (-x/9) + (5/9). This solution corresponds to option 3 in the provided options. It's crucial to understand what this inverse function represents and its relationship to the original function. The inverse function essentially reverses the operation of the original function. If we input a value into f(x) and then input the result into f⁻¹(x), we should obtain the original input value. This is a fundamental property of inverse functions.

To further illustrate this, let's consider an example. If we input x = 2 into f(x) = 5 - 9x, we get f(2) = 5 - 9(2) = 5 - 18 = -13. Now, if we input -13 into our inverse function f⁻¹(x) = (-x/9) + (5/9), we get f⁻¹(-13) = (-(-13)/9) + (5/9) = (13/9) + (5/9) = 18/9 = 2, which is our original input value. This confirms that our calculated inverse function is indeed correct. Furthermore, the graphs of f(x) and f⁻¹(x) are reflections of each other across the line y = x. This graphical representation provides another way to visualize the relationship between a function and its inverse. Understanding these implications deepens our comprehension of inverse functions and their role in mathematics.

Conclusion

In conclusion, we have successfully determined the inverse function for f(x) = 5 - 9x, which is f⁻¹(x) = (-x/9) + (5/9). This process involved replacing f(x) with y, swapping x and y, solving for y, and finally replacing y with f⁻¹(x). Understanding the concept of inverse functions is essential in mathematics, as they play a crucial role in various applications, including solving equations and analyzing relationships between functions. The inverse function essentially reverses the operation of the original function, and this property is fundamental to its definition.

The steps outlined in this article provide a systematic approach to finding the inverse of a function. By following these steps carefully, we can accurately determine the inverse for a wide range of functions. Moreover, understanding the implications of the inverse function, such as its relationship to the original function and its graphical representation, enhances our overall mathematical comprehension. The ability to find inverse functions is a valuable skill in many areas of mathematics and related fields, making this a crucial topic to master. This exploration of inverse functions demonstrates the interconnectedness of mathematical concepts and the importance of a solid foundation in fundamental principles.