Finding The Inverse Of F(x) = 1/(x+9) A Step-by-Step Guide

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In mathematics, finding the inverse of a function is a fundamental concept with applications across various fields. The inverse function, denoted as f⁻¹(x), essentially reverses the operation of the original function, f(x). In simpler terms, if f(a) = b, then f⁻¹(b) = a. This article provides a comprehensive guide on how to determine the inverse of a function, using the example of f(x) = 1/(x + 9) to illustrate the process. We will delve into the step-by-step methodology, the underlying principles, and the importance of understanding domain and range considerations when dealing with inverse functions. Understanding inverse functions is crucial not only for advanced mathematical concepts but also for practical applications in areas such as computer science, engineering, and economics. So, let's embark on this journey of unraveling the mysteries of inverse functions!

Understanding Inverse Functions

Before diving into the specific example of f(x) = 1/(x + 9), it's essential to grasp the core concept of inverse functions. An inverse function essentially “undoes” what the original function does. If we input a value x into a function f, and the output is y, then inputting y into the inverse function f⁻¹ should yield x. Mathematically, this relationship is expressed as f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. 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 is one where each input maps to a unique output, and conversely, each output corresponds to a unique input. Graphically, a function is one-to-one if it passes the horizontal line test, meaning no horizontal line intersects the graph of the function more than once. This property is crucial because it ensures that the inverse function will also be a function, meaning it will have a single, well-defined output for each input. The process of finding an inverse function involves a series of algebraic manipulations, which we will explore in detail in the subsequent sections. These manipulations aim to isolate the input variable in terms of the output variable, effectively reversing the functional relationship. Understanding the domain and range of both the original function and its inverse is also paramount, as these sets of values define the permissible inputs and outputs for the functions.

Step-by-Step Method to Find the Inverse Function

To find the inverse of a function, we follow a systematic approach that involves several key steps. Let's illustrate these steps using the example function f(x) = 1/(x + 9). The first step is to replace f(x) with y. This substitution makes the equation easier to manipulate algebraically. So, we rewrite the function as y = 1/(x + 9). The next crucial step is to swap x and y. This effectively reverses the roles of the input and output variables, setting the stage for finding the inverse relationship. After swapping, our equation becomes x = 1/(y + 9). Now comes the core algebraic manipulation: solving for y. This involves isolating y on one side of the equation. To do this, we can first multiply both sides of the equation by (y + 9), which gives us x(y + 9) = 1. Next, we distribute the x on the left side, resulting in xy + 9x = 1. Our goal is to isolate y, so we subtract 9x from both sides, obtaining xy = 1 - 9x. Finally, we divide both sides by x to solve for y, giving us y = (1 - 9x)/ x. The last step is to replace y with f⁻¹(x). This notation signifies that we have found the inverse function. Therefore, the inverse function of f(x) = 1/(x + 9) is f⁻¹(x) = (1 - 9x)/ x. It's important to note that this process may involve different algebraic techniques depending on the complexity of the original function, but the fundamental principle of swapping variables and solving for y remains the same. We will further explore the implications of this inverse function, including its domain and range, in the subsequent sections.

Applying the Method to f(x) = 1/(x+9)

Now, let's meticulously apply the step-by-step method we discussed earlier to the function f(x) = 1/(x + 9) to determine its inverse. This will provide a concrete example of how the process works in practice. As the first step, we replace f(x) with y, transforming the equation into y = 1/(x + 9). This substitution simplifies the algebraic manipulations that follow. Next, we swap x and y, effectively reversing the roles of the input and output variables. This crucial step sets the stage for finding the inverse relationship. After swapping, our equation becomes x = 1/(y + 9). The next stage involves solving for y. To achieve this, we first multiply both sides of the equation by (y + 9). This eliminates the fraction on the right side and gives us x(y + 9) = 1. We then distribute the x on the left side, resulting in xy + 9x = 1. Now, we aim to isolate y. We subtract 9x from both sides of the equation, which yields xy = 1 - 9x. Finally, we divide both sides by x to solve for y, giving us y = (1 - 9x)/ x. The final step in the process is to replace y with f⁻¹(x). This notation signifies that we have successfully found the inverse function. Therefore, the inverse function of f(x) = 1/(x + 9) is f⁻¹(x) = (1 - 9x)/ x. This result demonstrates the practical application of the method and highlights the algebraic steps involved in finding the inverse of a function. In the following sections, we will discuss how to verify this result and analyze the domain and range of both the original function and its inverse.

Verifying the Inverse Function

Once we've found a potential inverse function, it's crucial to verify that it is indeed the correct inverse. This verification process involves demonstrating that f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. These two conditions must hold true for the functions to be inverses of each other. Let's apply this verification process to our example, where f(x) = 1/(x + 9) and f⁻¹(x) = (1 - 9x)/ x. First, we'll evaluate f⁻¹(f(x)). This means we substitute f(x) into f⁻¹(x). So, we have f⁻¹(1/(x + 9)) = (1 - 9(1/(x + 9)))/(1/(x + 9)). To simplify this expression, we first multiply 9 by 1/(x + 9), which gives us 9/(x + 9). Then, the expression becomes (1 - 9/(x + 9))/(1/(x + 9)). To further simplify, we find a common denominator for the numerator, which is (x + 9). This gives us ((x + 9 - 9)/(x + 9))/(1/(x + 9)). Simplifying the numerator, we get (x/(x + 9))/(1/(x + 9)). Now, we divide the fractions by multiplying by the reciprocal of the denominator, which gives us (x/(x + 9)) * ((x + 9)/1). The (x + 9) terms cancel out, leaving us with x. Thus, f⁻¹(f(x)) = x. Next, we need to verify f(f⁻¹(x)). This means we substitute f⁻¹(x) into f(x). So, we have f((1 - 9x)/ x) = 1/(((1 - 9x)/ x) + 9). To simplify this expression, we first find a common denominator for the denominator of the main fraction, which is x. This gives us 1/(((1 - 9x) + 9x)/ x). Simplifying the denominator, we get 1/(1/ x). Dividing by a fraction is the same as multiplying by its reciprocal, so we have 1 * (x/1), which simplifies to x. Thus, f(f⁻¹(x)) = x. Since both f⁻¹(f(x)) = x and f(f⁻¹(x)) = x hold true, we have verified that f⁻¹(x) = (1 - 9x)/ x is indeed the inverse of f(x) = 1/(x + 9). This verification step is essential to ensure the correctness of the inverse function and provides confidence in the result.

Domain and Range of f(x) and f⁻¹(x)

Understanding the domain and range of a function and its inverse is crucial for a complete analysis. The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) that the function can produce. When dealing with inverse functions, the domain and range are closely related: the domain of the original function becomes the range of the inverse function, and the range of the original function becomes the domain of the inverse function. Let's analyze the domain and range of our example function, f(x) = 1/(x + 9), and its inverse, f⁻¹(x) = (1 - 9x)/ x. For f(x) = 1/(x + 9), the function is undefined when the denominator is equal to zero. This occurs when x + 9 = 0, which means x = -9. Therefore, the domain of f(x) is all real numbers except -9, which can be expressed as (-∞, -9) ∪ (-9, ∞). To find the range of f(x), we consider the possible output values. Since the numerator is 1, the function can take any value except 0. As x approaches -9, the function approaches either positive or negative infinity. Therefore, the range of f(x) is all real numbers except 0, which can be expressed as (-∞, 0) ∪ (0, ∞). Now, let's consider the inverse function, f⁻¹(x) = (1 - 9x)/ x. The function is undefined when the denominator is equal to zero, which occurs when x = 0. Therefore, the domain of f⁻¹(x) is all real numbers except 0, which can be expressed as (-∞, 0) ∪ (0, ∞). The range of f⁻¹(x) corresponds to the domain of the original function, f(x). Thus, the range of f⁻¹(x) is all real numbers except -9, which can be expressed as (-∞, -9) ∪ (-9, ∞). This analysis demonstrates the reciprocal relationship between the domains and ranges of a function and its inverse. Understanding these concepts is essential for accurately interpreting the behavior of functions and their inverses.

Conclusion

In conclusion, finding the inverse of a function is a crucial concept in mathematics with far-reaching applications. This article has provided a detailed guide on how to determine the inverse of a function, using the example of f(x) = 1/(x + 9) as an illustrative case. We've walked through the step-by-step methodology, emphasizing the importance of replacing f(x) with y, swapping x and y, solving for y, and replacing y with f⁻¹(x). The algebraic manipulations involved in this process are fundamental and require a solid understanding of equation solving. Furthermore, we've highlighted the significance of verifying the inverse function by demonstrating that f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. This verification step ensures the correctness of the derived inverse function. The discussion on the domain and range of both the original function and its inverse underscores the importance of considering these sets of values for a complete analysis. The domain of the original function becomes the range of the inverse, and vice versa, reflecting the inverse relationship between the functions. By understanding these concepts, we gain a deeper appreciation for the nature of functions and their inverses. The ability to find and verify inverse functions is not only essential for advanced mathematical studies but also for practical applications in various fields, including computer science, engineering, and economics. Therefore, mastering this skill is a valuable asset for anyone pursuing quantitative disciplines. The specific example of f(x) = 1/(x + 9) serves as a concrete illustration of the general method, providing a solid foundation for tackling more complex functions. As you continue your mathematical journey, remember the principles and techniques discussed in this article, and you'll be well-equipped to handle the challenges of inverse functions and related concepts.