Finding The Inverse Of F(x) = 64x^3 - 1 A Step-by-Step Solution
In mathematics, the inverse of a function, denoted as $f^{-1}(x)$, essentially undoes what the original function $f(x)$ does. In simpler terms, if $f(a) = b$, then $f^{-1}(b) = a$. Finding the inverse of a function is a crucial skill in algebra and calculus, enabling us to solve equations, analyze relationships between variables, and understand the fundamental properties of functions. In this comprehensive guide, we will delve into the process of finding the inverse of a function, specifically focusing on the example $f(x) = 64x^3 - 1$. We will break down each step, providing clear explanations and justifications to ensure a thorough understanding of the concept. Understanding inverse functions is crucial in various mathematical contexts, from solving equations to analyzing transformations. The inverse function, denoted as $f^{-1}(x)$, essentially “undoes” the operation of the original function $f(x)$. In this article, we will explore a step-by-step approach to finding the inverse of the given function, providing clear explanations and examples to help you grasp the underlying concepts. We'll begin by understanding the fundamental concept of inverse functions and then proceed with a methodical approach to find the inverse of $f(x) = 64x^3 - 1$. This involves swapping variables, solving for the new dependent variable, and expressing the result in the standard inverse function notation.
Step 1: Replace $f(x)$ with $y$
The first step in finding the inverse is to replace the function notation $f(x)$ with the variable $y$. This simple substitution helps to visualize the relationship between the input and output variables. By replacing $f(x)$ with $y$, we are essentially rewriting the function in a more convenient form for the subsequent steps. This is a standard practice in finding inverses and simplifies the algebraic manipulations involved. Our function, $f(x) = 64x^3 - 1$, now becomes: $y = 64x^3 - 1$. This transformation sets the stage for the next crucial step: swapping the variables $x$ and $y$. This foundational step allows us to manipulate the equation to solve for the inverse function. It's crucial to remember that this substitution is purely notational and doesn't change the function's behavior; it simply makes the process of finding the inverse more manageable. The key idea here is to prepare the equation for the variable swap that will define the inverse relationship. This substitution is not just a matter of convention; it's a deliberate step that simplifies the subsequent algebraic manipulations. By replacing $f(x)$ with $y$, we create a more symmetrical form that facilitates the swapping of variables, a critical step in finding the inverse function.
Step 2: Swap $x$ and $y$
This is the core step in finding the inverse function. We interchange the positions of $x$ and $y$ in the equation. This reflects the fundamental concept of an inverse function: it reverses the roles of input and output. In other words, if the original function takes $x$ as input and produces $y$ as output, the inverse function takes $y$ as input and produces $x$ as output. The act of swapping $x$ and $y$ mathematically embodies this reversal. Swapping the variables, $x$ and $y$, is the heart of finding the inverse. It reflects the very definition of an inverse function – reversing the roles of input and output. Where the original function takes an $x$ value and produces a $y$ value, the inverse function takes that $y$ value and returns the original $x$. In our equation, $y = 64x^3 - 1$, swapping $x$ and $y$ gives us: $x = 64y^3 - 1$. Now, our goal is to isolate $y$ on one side of the equation. This process of swapping variables is not just a mechanical step; it's a conceptual shift that reflects the fundamental nature of inverse functions. By interchanging $x$ and $y$, we are essentially looking at the function from the perspective of its inverse, setting the stage for solving for the inverse function explicitly. This step is essential because it directly implements the definition of an inverse function: reversing the roles of input and output. It allows us to rewrite the equation in a form where we can isolate $y$, which will then represent the inverse function, $f^{-1}(x)$.
Step 3: Solve for $y$
Now, we need to isolate $y$ in the equation $x = 64y^3 - 1$. This involves a series of algebraic manipulations. First, we add 1 to both sides of the equation to get: $x + 1 = 64y^3$. Next, we divide both sides by 64: $\fracx + 1}{64} = y^3$. Finally, to isolate $y$, we take the cube root of both sides64}}$. We can simplify this further by recognizing that 64 is a perfect cube ($4^3 = 64$), so we can rewrite the equation as}{4}$. The goal here is to express $y$ as a function of $x$, which will represent the inverse function. Solving for $y$ is a crucial step in the process. It involves using algebraic techniques to isolate $y$ on one side of the equation. Starting from $x = 64y^3 - 1$, we need to undo the operations that are being applied to $y$. This is done by performing the inverse operations in reverse order. First, we add 1 to both sides, then divide by 64, and finally take the cube root to isolate $y$. This process mirrors the steps of solving a regular algebraic equation, but with the added understanding that we are aiming to express the inverse relationship. Each algebraic manipulation must be performed carefully to maintain the equality and ensure that we arrive at the correct expression for $y$. This is a critical step, as any errors in the algebraic manipulations will lead to an incorrect inverse function. The final expression for $y$ will be the explicit form of the inverse function, allowing us to calculate the inverse for any given value of $x$.
Step 4: Replace $y$ with $f^{-1}(x)$
The last step is to replace $y$ with the inverse function notation, $f^-1}(x)$. This clearly indicates that we have found the inverse of the original function. So, our inverse function is(x) = \frac{\sqrt[3]{x + 1}}{4}$. This notation is essential for communicating that we are dealing with the inverse function and helps to distinguish it from the original function. This final step provides the formal representation of the inverse function. Replacing $y$ with $f^{-1}(x)$ is a matter of notation, but it's crucial for clarity and communication. It signifies that we have successfully found the inverse function and allows us to express it in standard mathematical notation. The notation $f^{-1}(x)$ is universally understood to represent the inverse of the function $f(x)$, making it essential for mathematical discourse. This replacement is not just a formality; it's a clear indication that we have completed the process of finding the inverse and are presenting the result in its proper form. This ensures that anyone reading our solution will immediately recognize it as the inverse of the original function. The use of the correct notation is vital for avoiding confusion and maintaining mathematical rigor.
Conclusion
Therefore, the inverse of the function $f(x) = 64x^3 - 1$ is $f^{-1}(x) = \frac{\sqrt[3]{x + 1}}{4}$. This corresponds to option C. Finding the inverse of a function is a systematic process that involves understanding the fundamental concepts and applying algebraic manipulations carefully. By following these steps, you can confidently find the inverse of various functions. In this comprehensive guide, we've meticulously walked through each step of finding the inverse of $f(x) = 64x^3 - 1$, emphasizing the importance of each step and the underlying principles of inverse functions. The ability to find the inverse of a function is a cornerstone of advanced mathematical concepts and is essential for a deep understanding of functions and their properties. The process of finding the inverse not only tests algebraic skills but also reinforces the conceptual understanding of how functions and their inverses relate to each other. By mastering this process, you will be well-equipped to tackle more complex mathematical problems and gain a deeper appreciation for the beauty and logic of mathematics. We've demonstrated how to systematically approach the problem, breaking it down into manageable steps and providing clear explanations along the way. This approach can be applied to finding the inverses of a wide range of functions, making it a valuable skill for anyone studying mathematics.
Correct Answer
C. $f^{-1}(x)=\frac{\sqrt[3]{x+1}}{4}$
