Finding The Inverse Function F Inverse Of X When F Of X Equals 4x-5 Over X+5

In the realm of mathematics, the concept of inverse functions holds a significant position. An inverse function, denoted as $f^{-1}(x)$, essentially reverses the operation performed by the original function, $f(x)$. In simpler terms, if $f(a) = b$, then $f^{-1}(b) = a$. The purpose of this comprehensive guide is to provide a step-by-step explanation of how to determine the inverse function, $f^{-1}(x)$, when given the function $f(x) = \frac{4x - 5}{x + 5}$. This process involves a series of algebraic manipulations aimed at isolating $x$ in terms of $y$, and subsequently swapping the variables to express the inverse function in the standard form. Mastering this technique is crucial for students and professionals alike, as inverse functions appear in various branches of mathematics and its applications, including calculus, algebra, and complex analysis.

Understanding Inverse Functions

Before diving into the specific example, let's solidify our understanding of inverse functions. An inverse function exists if and only if the original function is one-to-one, meaning that it passes both the horizontal and vertical line tests. A one-to-one function ensures that each input value maps to a unique output value, and vice versa. Graphically, this implies that no horizontal line intersects the graph of the function more than once. The inverse function essentially "undoes" the operation of the original function. If we apply a function $f$ to a value $x$ and then apply its inverse $f^{-1}$ to the result, we should obtain the original value $x$. Mathematically, this is expressed as $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$. These identities are fundamental in verifying whether a function and its proposed inverse are indeed inverses of each other. The process of finding an inverse function involves several key steps, including replacing $f(x)$ with $y$, swapping $x$ and $y$, and then solving for $y$. This resulting expression for $y$ represents the inverse function, $f^{-1}(x)$. Understanding these core concepts is essential for successfully navigating the steps involved in finding the inverse of a function, particularly in cases involving rational functions like the one we will explore.

Step 1: Replace $f(x)$ with $y$

The initial step in finding the inverse function is to replace the notation $f(x)$ with the variable $y$. This substitution simplifies the equation and makes it easier to manipulate algebraically. In our case, the given function is $f(x) = \frac{4x - 5}{x + 5}$. Replacing $f(x)$ with $y$, we obtain the equation:

$y = \frac{4x - 5}{x + 5}$

This seemingly simple substitution is a crucial step as it allows us to treat the function as an algebraic equation with two variables, $x$ and $y$. This form is more amenable to the algebraic manipulations required to isolate $x$ in terms of $y$, which is the core of finding the inverse function. By replacing $f(x)$ with $y$, we set the stage for the subsequent steps, which involve swapping the variables and solving for $y$. This step is not merely a cosmetic change; it fundamentally transforms the way we view the function, shifting our focus from expressing $y$ in terms of $x$ to expressing $x$ in terms of $y$. This change in perspective is essential for understanding the concept of an inverse function as a reversal of the original function's operation. The ease with which we can manipulate the equation in this form is a testament to the importance of this initial substitution. Furthermore, this step aligns with the general procedure for finding inverses of various types of functions, making it a fundamental technique to master.

Step 2: Swap $x$ and $y$

Following the replacement of $f(x)$ with $y$, the next pivotal step in determining the inverse function is to swap the variables $x$ and $y$. This step embodies the core concept of an inverse function, which is to reverse the roles of input and output. Starting with the equation obtained in the previous step, $y = \frac{4x - 5}{x + 5}$, we interchange $x$ and $y$ to get:

$x = \frac{4y - 5}{y + 5}$

This swapping of variables is not just a mechanical procedure; it reflects the fundamental idea that the inverse function maps the output of the original function back to its input. In other words, if the original function takes $x$ to $y$, the inverse function takes $y$ back to $x$. This step is crucial because it sets up the equation that we will solve for $y$ to obtain the expression for the inverse function. By interchanging $x$ and $y$, we are essentially changing our perspective from expressing $y$ as a function of $x$ to expressing $x$ as a function of $y$. This shift is necessary to find the inverse function, which expresses $y$ as a function of $x$, but in a way that "undoes" the original function. The simplicity of this step belies its profound significance in the process of finding inverse functions. It is the key to reversing the operation of the original function and obtaining the desired inverse. Moreover, this step is universally applicable to finding inverses of various functions, making it a cornerstone technique in mathematical manipulations.

Step 3: Solve for $y$

After swapping $x$ and $y$, the crucial next step in finding the inverse function is to solve the resulting equation for $y$. This involves a series of algebraic manipulations aimed at isolating $y$ on one side of the equation. Starting with the equation obtained in the previous step, $x = \frac{4y - 5}{y + 5}$, we will now solve for $y$. First, we multiply both sides of the equation by $(y + 5)$ to eliminate the denominator:

$x(y + 5) = 4y - 5$

Next, we distribute $x$ on the left side of the equation:

$xy + 5x = 4y - 5$

Now, we want to group all terms containing $y$ on one side of the equation and all other terms on the other side. To do this, we subtract $4y$ from both sides and subtract $5x$ from both sides:

$xy - 4y = -5x - 5$

We can now factor out $y$ from the left side of the equation:

$y(x - 4) = -5x - 5$

Finally, we divide both sides by $(x - 4)$ to isolate $y$:

$y = \frac{-5x - 5}{x - 4}$

This expression for $y$ is the key to finding the inverse function. Solving for $y$ is often the most challenging part of the process, as it may involve various algebraic techniques depending on the complexity of the function. However, by systematically applying the principles of algebraic manipulation, we can successfully isolate $y$ and express it in terms of $x$. This step is the culmination of the previous steps, as it provides us with the explicit form of the inverse function. The ability to confidently solve for $y$ is a testament to one's algebraic skills and is crucial for mastering the concept of inverse functions. Furthermore, this step highlights the importance of careful and methodical manipulation to avoid errors and arrive at the correct solution.

Step 4: Replace $y$ with $f^{-1}(x)$

The final step in determining the inverse function is to replace the variable $y$ with the notation $f^{-1}(x)$. This step is crucial for expressing the inverse function in standard mathematical notation. From the previous step, we obtained the expression $y = \frac{-5x - 5}{x - 4}$. Replacing $y$ with $f^{-1}(x)$, we get:

$f^{-1}(x) = \frac{-5x - 5}{x - 4}$

This is the inverse function of the original function $f(x) = \frac{4x - 5}{x + 5}$. The notation $f^{-1}(x)$ explicitly denotes the inverse function, making it clear that this function reverses the operation of the original function $f(x)$. This final step is not just a matter of notation; it signifies the completion of the process of finding the inverse function. By replacing $y$ with $f^{-1}(x)$, we are formally expressing the inverse relationship between the input and output of the original function. This notation is universally recognized and understood in mathematics, making it essential for clear communication and understanding. Furthermore, this step underscores the importance of using proper mathematical notation to convey concepts accurately and effectively. The ability to correctly express the inverse function using the $f^{-1}(x)$ notation demonstrates a solid understanding of the concept and its representation. This final substitution is the finishing touch that transforms the algebraic expression into a formal mathematical representation of the inverse function.

Verification (Optional but Recommended)

While we have successfully derived the inverse function, it is always a prudent practice to verify our result. Verification ensures that the function we have found truly reverses the operation of the original function. To verify the inverse, we need to show that both $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Let's perform these checks:

First, let's compute $f(f^{-1}(x))$:

$f(f^{-1}(x)) = f\left(\frac{-5x - 5}{x - 4}\right) = \frac{4\left(\frac{-5x - 5}{x - 4}\right) - 5}{\left(\frac{-5x - 5}{x - 4}\right) + 5}$

To simplify this expression, we multiply the numerator and denominator by $(x - 4)$:

$= \frac{4(-5x - 5) - 5(x - 4)}{(-5x - 5) + 5(x - 4)} = \frac{-20x - 20 - 5x + 20}{-5x - 5 + 5x - 20} = \frac{-25x}{-25} = x$

Thus, $f(f^{-1}(x)) = x$.

Now, let's compute $f^{-1}(f(x))$:

$f^{-1}(f(x)) = f^{-1}\left(\frac{4x - 5}{x + 5}\right) = \frac{-5\left(\frac{4x - 5}{x + 5}\right) - 5}{\left(\frac{4x - 5}{x + 5}\right) - 4}$

Again, we multiply the numerator and denominator by $(x + 5)$:

$= \frac{-5(4x - 5) - 5(x + 5)}{(4x - 5) - 4(x + 5)} = \frac{-20x + 25 - 5x - 25}{4x - 5 - 4x - 20} = \frac{-25x}{-25} = x$

Thus, $f^{-1}(f(x)) = x$.

Since both $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$, we have verified that our inverse function is correct. This verification step is not just a formality; it provides a crucial check against errors that may have occurred during the algebraic manipulations. By performing this verification, we can be confident that the function we have found truly reverses the operation of the original function. This rigorous approach ensures the accuracy of our result and reinforces our understanding of the concept of inverse functions. The verification process also highlights the symmetry inherent in inverse functions, as they essentially "undo" each other. This understanding is essential for applying inverse functions in various mathematical contexts and problem-solving scenarios.

Conclusion

In summary, finding the inverse function $f^{-1}(x)$ for a given function $f(x)$, such as $f(x) = \frac{4x - 5}{x + 5}$, involves a systematic process of algebraic manipulation. The key steps include replacing $f(x)$ with $y$, swapping $x$ and $y$, solving for $y$, and finally, replacing $y$ with $f^{-1}(x)$. This step-by-step approach allows us to reverse the operation of the original function and express the inverse relationship in standard mathematical notation. While the algebraic manipulations may vary depending on the complexity of the function, the underlying principle remains the same: to isolate $y$ in terms of $x$ after swapping the variables. Furthermore, the optional but highly recommended verification step provides a crucial check against errors and reinforces our understanding of the concept of inverse functions. By demonstrating that both $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$, we can be confident in the accuracy of our result. Mastering the technique of finding inverse functions is essential for success in various branches of mathematics and its applications. This skill not only enhances our problem-solving abilities but also deepens our understanding of the fundamental relationships between functions and their inverses. The process outlined in this guide provides a solid foundation for tackling more complex inverse function problems and applying this knowledge in diverse mathematical contexts. From calculus to cryptography, the concept of inverse functions plays a vital role, making it a cornerstone of mathematical literacy.