Finding The Inverse Of F(x) = ³√(x + 12) A Step-by-Step Guide

In mathematics, the inverse of a function, often 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. This concept is fundamental in various areas of mathematics, including algebra, calculus, and analysis. To find the inverse of a function, we typically swap the roles of x and y and then solve for y. This process effectively reverses the input-output relationship of the original function. The inverse function exists only if the original function is one-to-one, meaning that each input value corresponds to a unique output value, and vice versa. Graphically, a function is one-to-one if it passes the horizontal line test, where no horizontal line intersects the graph more than once. Understanding inverse functions is crucial for solving equations, simplifying expressions, and grasping more advanced mathematical concepts. For instance, in calculus, inverse functions play a significant role in integration and differentiation. Moreover, in real-world applications, inverse functions are used in cryptography, data analysis, and various engineering disciplines. The process of finding an inverse involves algebraic manipulation and a solid understanding of function properties. This skill is not only essential for academic success in mathematics but also for practical problem-solving in diverse fields. Therefore, mastering the concept of inverse functions is a valuable investment in one's mathematical toolkit.

The given function is:

$$f(x) = \sqrt[3]{x + 12}$$

We are tasked with finding the inverse of this function, denoted as f⁻¹(x). This involves a series of algebraic steps to reverse the operations performed by the original function. The function f(x) first adds 12 to the input x, and then takes the cube root of the result. To find the inverse, we need to reverse these operations in the opposite order. This means we will first undo the cube root by cubing, and then undo the addition of 12 by subtracting 12. The process of finding an inverse function is a fundamental concept in algebra and calculus, as it allows us to reverse the relationship between input and output. This is particularly useful in solving equations and understanding the behavior of functions. The correct inverse function will satisfy the condition that f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. These conditions ensure that the inverse function truly reverses the effect of the original function. Therefore, the task at hand is to carefully apply the steps of swapping variables and solving for y to arrive at the correct inverse function. This requires a clear understanding of algebraic manipulation and the properties of cube roots and exponents. The following steps will detail the process of finding the inverse, explaining each step in a clear and concise manner.

To find the inverse of the function f(x) = ³√(x + 12), we follow these steps:

1. Replace f(x) with y:

   This gives us:

   $$y = \sqrt[3]{x + 12}$$

   Replacing f(x) with y is a standard first step in finding the inverse of a function. It simplifies the notation and makes it easier to manipulate the equation. The equation y = ³√(x + 12) represents the same relationship as the original function, but now in a form that is more conducive to finding the inverse. This step is crucial because it sets the stage for the next step, which involves swapping x and y. By making this substitution, we are preparing the equation to be rearranged in terms of the new y, which will represent the inverse function. The use of y as a placeholder for f(x) is a common practice in mathematics, especially when dealing with functions and their inverses. It allows us to treat the function's output as a variable that can be manipulated algebraically. This substitution is not just a matter of notation; it is a conceptual shift that allows us to view the function from a different perspective, one that is essential for understanding and finding its inverse.

2. Swap x and y:

   Interchange x and y to get:

   $$x = \sqrt[3]{y + 12}$$

   Swapping x and y is the heart of finding the inverse function. This step reflects the fundamental idea that the inverse function reverses the roles of input and output. What was once the input (x) becomes the output (y), and vice versa. The equation x = ³√(y + 12) now represents the inverse relationship, but it is not yet in the standard form of a function, where y is expressed in terms of x. This swapping step is a critical transformation that captures the essence of the inverse function. It is not just a mechanical procedure; it is a conceptual move that acknowledges the reversed mapping between the domain and range of the original function and its inverse. The swapped equation holds the key to finding the inverse, but it requires further algebraic manipulation to isolate y and express it as a function of x. This step is crucial for understanding the relationship between a function and its inverse, as it visually demonstrates how the roles of input and output are interchanged.

3. Solve for y:

   To isolate y, first, cube both sides of the equation:

   $$x^3 = (\sqrt[3]{y + 12})^3$$

   This simplifies to:

   $$x^3 = y + 12$$

   Next, subtract 12 from both sides:

   $$x^3 - 12 = y$$

   Solving for y involves a series of algebraic manipulations aimed at isolating y on one side of the equation. In this case, the first step is to eliminate the cube root by cubing both sides of the equation. This is a fundamental algebraic operation that preserves the equality while simplifying the equation. The equation x³ = y + 12 now represents the inverse relationship without the cube root, making it easier to isolate y. The next step is to subtract 12 from both sides, which is another algebraic operation that maintains equality. This step isolates y completely, expressing it as a function of x. The equation x³ - 12 = y is the inverse function in its explicit form, where y is clearly defined in terms of x. This process of solving for y demonstrates the power of algebraic manipulation in transforming equations and revealing hidden relationships. Each step is carefully chosen to undo the operations performed on y, ultimately leading to its isolation.

4. Replace y with f⁻¹(x)*:

   Finally, replace y with f⁻¹(x) to denote the inverse function:

   $$f^{-1}(x) = x^3 - 12$$

   Replacing y with f⁻¹(x) is the final step in expressing the inverse function in standard notation. This notation clearly indicates that the function we have found is the inverse of the original function f(x). The equation f⁻¹(x) = x³ - 12 represents the inverse function in a concise and recognizable form. This step is crucial for clarity and consistency in mathematical communication. By using the notation f⁻¹(x), we explicitly state that we have found the function that reverses the operation of f(x). This notation is widely used in mathematics and is essential for understanding and working with inverse functions. The final result, f⁻¹(x) = x³ - 12, is the answer to the problem and represents the inverse of the original function f(x) = ³√(x + 12). This step completes the process of finding the inverse, and the result can be used for further analysis or application.

The inverse of the function f(x) = ³√(x + 12) is:

$$f^{-1}(x) = x^3 - 12$$

Therefore, the correct answer is A. f⁻¹(x) = x³ - 12. This solution has been derived through a step-by-step process that involves swapping the variables x and y and then solving for y. This method is a standard approach for finding the inverse of a function and ensures that the resulting function f⁻¹(x) indeed reverses the operation of the original function f(x). The verification of this answer can be done by checking if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. These conditions are essential to confirm that the inverse function is correctly found. The correct answer, f⁻¹(x) = x³ - 12, demonstrates the relationship between the cube root function and the cubic function, highlighting how they are inverses of each other. Understanding this relationship is crucial in various mathematical contexts, including solving equations, simplifying expressions, and analyzing function behavior. Therefore, the correct identification of the inverse function is a significant accomplishment in mastering the concept of inverse functions.

Let's analyze why the other options are incorrect:

• B. f⁻¹(x) = x - 12: This option only subtracts 12 from x and does not account for the cube root in the original function. It fails to reverse the cubing operation, which is essential for finding the inverse.

• C. f⁻¹(x) = x + 12: This option incorrectly adds 12 to x instead of accounting for the cube root. It only addresses the addition within the original function but misses the crucial step of reversing the cube root operation.

• D. f⁻¹(x) = 12 - x³: This option has the correct components (x³ and 12) but the subtraction is in the wrong order. It reverses the cubing operation but then subtracts x³ from 12 instead of subtracting 12 from x³.

Each of these incorrect options demonstrates a misunderstanding of the process of finding an inverse function. Option B only addresses the addition of 12 but neglects the cube root. Option C also fails to account for the cube root and performs the addition in the wrong direction. Option D includes the correct operations but reverses the order of subtraction, leading to an incorrect result. These errors highlight the importance of carefully following the steps of swapping variables and solving for y to correctly find the inverse function. The analysis of these incorrect options provides valuable insight into common mistakes and reinforces the correct method for finding inverse functions. Understanding why these options are wrong is as important as understanding why the correct answer is right, as it deepens the understanding of the underlying concepts and processes.

In conclusion, the inverse of the function f(x) = ³√(x + 12) is f⁻¹(x) = x³ - 12. This was determined by following the standard procedure for finding inverse functions: replacing f(x) with y, swapping x and y, solving for y, and then replacing y with f⁻¹(x). The correct answer, option A, demonstrates a clear understanding of how to reverse the operations of the original function. The incorrect options highlight common mistakes, such as failing to account for all operations or performing them in the wrong order. This problem underscores the importance of mastering the concept of inverse functions, as it is a fundamental topic in algebra and calculus. The ability to find inverse functions is crucial for solving equations, simplifying expressions, and understanding the relationships between different functions. The step-by-step solution provided in this discussion offers a clear and concise method for finding inverse functions, and the analysis of incorrect options reinforces the importance of careful attention to detail and a thorough understanding of algebraic principles. This problem serves as a valuable exercise in mathematical reasoning and problem-solving, enhancing one's overall mathematical proficiency.