Finding The Inverse Function Of F(x) = 1/(x+4)

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In mathematics, inverse functions play a crucial role in understanding the relationship between functions and their reversed operations. Given a function f(x), its inverse, denoted as f-1(x), essentially "undoes" what the original function does. In simpler terms, if f(a) = b, then f-1(b) = a. Finding the inverse of a function is a fundamental concept in algebra and calculus, and it has practical applications in various fields, including cryptography, computer science, and engineering. In this article, we will delve into the process of finding the inverse of the function f(x) = 1/(x+4), providing a step-by-step guide and explaining the underlying principles.

To truly grasp the concept of inverse functions, it's vital to first understand what a function is and how it operates. A function can be thought of as a machine that takes an input, processes it according to a specific rule, and produces an output. The rule that defines the function dictates the relationship between the input and the output. For example, in the function f(x) = 1/(x+4), the input x is first added to 4, and then the reciprocal of the result is taken. The inverse function, f-1(x), reverses this process. It takes the output of the original function as its input and produces the original input as its output. This reversal is achieved by systematically undoing each operation performed by the original function in the reverse order. The process involves swapping the roles of x and y (where y = f(x)) and then solving for y. This resulting expression will be the inverse function, f-1(x). Understanding this concept is crucial for solving various mathematical problems and gaining a deeper insight into the behavior of functions.

Before we dive into the step-by-step process, it’s important to consider the domain and range of both the original function and its inverse. The domain of a function is the set of all possible input values (x-values) for which the function is defined, and the range is the set of all possible output values (y-values) that the function can produce. When finding the inverse, the domain of the original function becomes the range of the inverse, and the range of the original function becomes the domain of the inverse. This is a critical aspect to keep in mind, especially when dealing with functions that have restrictions on their domain, such as rational functions (like the one we're considering) or functions involving square roots or logarithms. For the function f(x) = 1/(x+4), the domain is all real numbers except x = -4, since division by zero is undefined. The range is all real numbers except y = 0, because the fraction can never equal zero. This understanding of domain and range will be crucial in verifying the correctness of the inverse function we find.

Step-by-Step Guide to Finding the Inverse

Finding the inverse of a function involves a systematic approach that ensures we correctly reverse the operations performed by the original function. The process typically consists of four main steps, each crucial to arriving at the correct inverse function. Let's walk through these steps for our function, f(x) = 1/(x+4).

Step 1: Replace f(x) with y

The first step in finding the inverse of a function is to replace the function notation f(x) with the variable y. This substitution makes the equation easier to manipulate algebraically. In essence, we are rewriting the function in a form that allows us to clearly see the relationship between the input x and the output y. For our function, f(x) = 1/(x+4), replacing f(x) with y gives us the equation:

y = 1/(x+4)

This simple substitution sets the stage for the subsequent steps, where we will be swapping the variables and solving for the new y. It's a fundamental step that helps in visualizing the function as a relationship between two variables rather than a function of a single variable.

Step 2: Swap x and y

The second, and arguably most crucial, step in finding the inverse function is to swap the positions of the variables x and y. This step reflects the fundamental idea of an inverse function – reversing the roles of input and output. By swapping x and y, we are essentially setting up the equation to solve for the inverse function, where the original output becomes the new input, and the original input becomes the new output. Applying this to our equation, y = 1/(x+4), we swap x and y to get:

x = 1/(y+4)

This equation now represents the inverse relationship, and the next step will involve solving for y to explicitly express the inverse function.

Step 3: Solve for y

The third step involves solving the equation obtained in Step 2 for y. This is where algebraic manipulation comes into play. Our goal is to isolate y on one side of the equation, expressing it in terms of x. This will give us the explicit form of the inverse function. Starting with our equation from Step 2, x = 1/(y+4), we can solve for y as follows:

  1. Multiply both sides of the equation by (y+4) to eliminate the fraction:
    x(y+4) = 1
  2. Distribute x on the left side: xy + 4x = 1
  3. Subtract 4x from both sides to isolate the term containing y: xy = 1 - 4x
  4. Divide both sides by x to solve for y: y = (1 - 4x) / x

This algebraic manipulation gives us y expressed in terms of x, which is a crucial step in determining the inverse function.

Step 4: Replace y with f-1(x)

The final step in finding the inverse function is to replace y with the inverse function notation, f-1(x). This notation explicitly indicates that the function we have found is the inverse of the original function f(x). Using the result from Step 3, y = (1 - 4x) / x, we replace y with f-1(x) to get:

f-1(x) = (1 - 4x) / x

This is the inverse function of f(x) = 1/(x+4). It represents the function that, when applied to the output of f(x), will return the original input. The notation f-1(x) is a standard way to denote the inverse function, making it clear that this function "undoes" the operation of the original function f(x). Now, we have successfully found the inverse function using the step-by-step method.

The Inverse Function: f-1(x) = (1 - 4x) / x

After following the step-by-step process, we have determined that the inverse function of f(x) = 1/(x+4) is:

f-1(x) = (1 - 4x) / x

This function, f-1(x), represents the reversed operation of the original function f(x). It's important to recognize that the inverse function effectively "undoes" what the original function does. If we input a value into f(x) and then input the result into f-1(x), we should get back our original input value. This property is a key characteristic of inverse functions and can be used to verify the correctness of our result. To ensure the accuracy of our derived inverse function, let's perform a verification step.

Verifying the Inverse Function

To verify that f-1(x) = (1 - 4x) / x is indeed the inverse of f(x) = 1/(x+4), we need to demonstrate that the composition of the two functions, in both orders, results in the identity function, x. In other words, we need to show that f(f-1(x)) = x and f-1(f(x)) = x. This verification process confirms that the inverse function correctly reverses the operation of the original function.

Verification 1: f(f-1(x)) = x

Let's start by computing f(f-1(x)). This means we will substitute f-1(x) into the original function f(x) wherever we see x.

f(f-1(x)) = f((1 - 4x) / x) = 1 / (((1 - 4x) / x) + 4)

Now, we simplify the expression:

  1. Find a common denominator to add the terms in the denominator: 1 / (((1 - 4x) / x) + (4x / x)) = 1 / ((1 - 4x + 4x) / x)
  2. Simplify the numerator in the denominator: 1 / (1 / x)
  3. Dividing by a fraction is the same as multiplying by its reciprocal: 1 * (x / 1) = x

Thus, f(f-1(x)) = x, which confirms the first part of our verification.

Verification 2: f-1(f(x)) = x

Next, we compute f-1(f(x)). This means we will substitute f(x) into the inverse function f-1(x) wherever we see x.

f-1(f(x)) = f-1(1/(x+4)) = (1 - 4(1/(x+4))) / (1/(x+4))

Now, we simplify the expression:

  1. Distribute the 4 in the numerator: (1 - (4/(x+4))) / (1/(x+4))
  2. Find a common denominator to subtract the terms in the numerator: (((x+4)/(x+4)) - (4/(x+4))) / (1/(x+4)) = ((x+4-4)/(x+4)) / (1/(x+4))
  3. Simplify the numerator: (x/(x+4)) / (1/(x+4))
  4. Dividing by a fraction is the same as multiplying by its reciprocal: (x/(x+4)) * ((x+4)/1) = x

Thus, f-1(f(x)) = x, which confirms the second part of our verification. Since both f(f-1(x)) = x and f-1(f(x)) = x, we can confidently conclude that f-1(x) = (1 - 4x) / x is indeed the inverse function of f(x) = 1/(x+4).

Domain and Range of the Inverse Function

When dealing with inverse functions, it's essential to consider the domain and range of both the original function and its inverse. As previously mentioned, the domain of the original function becomes the range of the inverse, and the range of the original function becomes the domain of the inverse. This interrelationship is crucial for a comprehensive understanding of how inverse functions operate.

For the original function, f(x) = 1/(x+4):

  • The domain is all real numbers except x = -4, because the denominator cannot be zero. In interval notation, this is (-∞, -4) ∪ (-4, ∞).
  • The range is all real numbers except y = 0, because the fraction can never equal zero. In interval notation, this is (-∞, 0) ∪ (0, ∞).

Now, let's consider the inverse function, f-1(x) = (1 - 4x) / x:

  • The domain is all real numbers except x = 0, because the denominator cannot be zero. In interval notation, this is (-∞, 0) ∪ (0, ∞).
  • The range is all real numbers except y = -4. To see this, we can rewrite the inverse function as f-1(x) = 1/x - 4. As x approaches infinity, 1/x approaches 0, so f-1(x) approaches -4, but never actually equals -4. In interval notation, this is (-∞, -4) ∪ (-4, ∞).

As expected, the domain of f(x) matches the range of f-1(x), and the range of f(x) matches the domain of f-1(x). This confirms our understanding of the relationship between the domain and range of a function and its inverse.

Graphical Representation of Inverse Functions

The graphical representation of inverse functions provides a visual way to understand their relationship. The graph of a function and its inverse are reflections of each other across the line y = x. This symmetry arises from the swapping of x and y coordinates when finding the inverse. If a point (a, b) lies on the graph of f(x), then the point (b, a) lies on the graph of f-1(x). This reflection property is a fundamental characteristic of inverse functions and can be a helpful tool for visualizing and verifying inverse relationships.

To illustrate this, consider the graphs of f(x) = 1/(x+4) and its inverse, f-1(x) = (1 - 4x) / x. The graph of f(x) is a hyperbola with a vertical asymptote at x = -4 and a horizontal asymptote at y = 0. The graph of f-1(x) is also a hyperbola, but with a vertical asymptote at x = 0 and a horizontal asymptote at y = -4. If you were to draw both graphs on the same coordinate plane along with the line y = x, you would see that they are mirror images of each other across this line. This visual confirmation reinforces the concept of inverse functions as reversed operations and provides an intuitive understanding of their symmetry.

Applications of Inverse Functions

Inverse functions are not just abstract mathematical concepts; they have practical applications in various fields. Understanding inverse functions can be crucial in solving real-world problems in areas such as cryptography, computer science, and engineering. One common application is in encoding and decoding information, where one function encodes a message, and its inverse decodes it. This is a fundamental principle behind many encryption algorithms.

In computer science, inverse functions are used in data compression and decompression. A compression algorithm reduces the size of a file, and its inverse, the decompression algorithm, restores the file to its original state. Similarly, in engineering, inverse functions can be used to solve equations and model systems. For example, if you have a function that describes the output of a system given an input, the inverse function can be used to determine the input required to achieve a desired output. These are just a few examples of how inverse functions play a vital role in various practical applications, highlighting their significance beyond theoretical mathematics.

Conclusion

In conclusion, finding the inverse of a function, such as f(x) = 1/(x+4), is a fundamental concept in mathematics with wide-ranging applications. The step-by-step process involves replacing f(x) with y, swapping x and y, solving for y, and replacing y with f-1(x). We successfully found that the inverse function of f(x) = 1/(x+4) is f-1(x) = (1 - 4x) / x. We verified this result by demonstrating that f(f-1(x)) = x and f-1(f(x)) = x. Understanding the domain and range of both the original function and its inverse is crucial, and we confirmed that the domain of f(x) matches the range of f-1(x), and vice versa. The graphical representation of inverse functions as reflections across the line y = x provides a visual understanding of their relationship.

Inverse functions have significant practical applications in fields such as cryptography, computer science, and engineering, making the understanding of this concept essential. By mastering the techniques for finding and verifying inverse functions, you can gain a deeper understanding of mathematical relationships and enhance your problem-solving skills in various contexts. The process of finding an inverse function not only reinforces algebraic manipulation skills but also provides valuable insights into the nature of functions and their reversed operations.