Finding The Inverse Function Of F(x) = (1/4)x - 12 A Step-by-Step Guide

Understanding Inverse Functions: A Foundation for Mathematical Exploration

In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. However, the concept of an inverse function introduces a fascinating twist, allowing us to reverse this mapping and trace outputs back to their original inputs. In essence, an inverse function acts as a mirror, reflecting the original function's operations in reverse order. This intricate dance between functions and their inverses holds profound significance in various mathematical disciplines, including calculus, algebra, and cryptography. To truly grasp the essence of inverse functions, we embark on a journey to unravel their definition, properties, and practical applications, focusing on the linear function f(x) = (1/4)x - 12 as a central example. Inverse functions play a crucial role in solving equations, simplifying complex expressions, and gaining deeper insights into the relationships between mathematical entities. The ability to find and manipulate inverse functions unlocks a powerful toolkit for problem-solving and mathematical exploration.

The journey into inverse functions begins with a clear understanding of their defining characteristic: they undo the action of the original function. Mathematically, if a function f maps an input x to an output y, then its inverse, denoted as f⁻¹ (y), maps y back to x. This seemingly simple concept has profound implications, as it allows us to reverse the flow of information and solve for unknown inputs given known outputs. For instance, if we know the output of a function and seek to find the input that produced it, the inverse function provides the direct pathway. This fundamental property underpins the usefulness of inverse functions in various applications, from solving equations to decoding encrypted messages. The concept of inverse functions is not merely an abstract mathematical curiosity; it is a practical tool that empowers us to manipulate and understand mathematical relationships with greater precision.

Deciphering the Inverse Function: A Step-by-Step Approach

The process of finding the inverse of a function involves a systematic approach, typically consisting of three key steps. First, we replace the function notation f(x) with the variable y, setting the stage for the algebraic manipulation that follows. Second, we interchange x and y, effectively reversing the roles of input and output. This step embodies the very essence of finding an inverse, as we are now expressing the original input as a function of the original output. Finally, we solve the resulting equation for y, isolating the variable that represents the inverse function. The expression we obtain after this step is the inverse function, denoted as f⁻¹(x). Applying this method to the linear function f(x) = (1/4)x - 12, we initially replace f(x) with y, yielding y = (1/4)x - 12. Subsequently, we interchange x and y, resulting in x = (1/4)y - 12. Our final step involves solving this equation for y. Adding 12 to both sides gives x + 12 = (1/4)y, and multiplying both sides by 4 yields y = 4x + 48. This expression represents the inverse function of f(x) = (1/4)x - 12. Finding the inverse function requires a careful execution of these steps, ensuring that each algebraic manipulation preserves the equality and leads us closer to the desired expression.

Unveiling the Inverse of f(x) = (1/4)x - 12: A Detailed Exploration

Let's embark on a step-by-step journey to determine the inverse of the function f(x) = (1/4)x - 12. This function represents a linear transformation, where the input x is first multiplied by 1/4 and then decreased by 12. To find the inverse, we aim to reverse these operations in the opposite order. The initial step involves replacing f(x) with y, giving us the equation y = (1/4)x - 12. This substitution simplifies the subsequent algebraic manipulations. Next, we interchange x and y, which is the core step in finding the inverse function. This swap reflects the fundamental idea of reversing the input-output relationship. The equation now becomes x = (1/4)y - 12. Our objective is to isolate y on one side of the equation. To achieve this, we first add 12 to both sides, resulting in x + 12 = (1/4)y. Then, we multiply both sides by 4 to eliminate the fraction, leading to 4(x + 12) = y. Distributing the 4 on the left side gives us y = 4x + 48. Therefore, the inverse function of f(x) = (1/4)x - 12 is h(x) = 4x + 48. The detailed steps involved in finding the inverse function highlight the importance of careful algebraic manipulation and a clear understanding of the underlying principles.

Verifying the Inverse: A Crucial Test of Accuracy

To ensure the correctness of our calculated inverse function, h(x) = 4x + 48, we can employ a crucial verification step. The defining property of inverse functions is that when they are composed with the original function, they should effectively cancel each other out, returning the original input. In other words, f(h(x)) should equal x, and h(f(x)) should also equal x. Let's first evaluate f(h(x)). Substituting h(x) into f(x), we get f(h(x)) = f(4x + 48) = (1/4)(4x + 48) - 12. Distributing the 1/4, we have f(h(x)) = x + 12 - 12, which simplifies to f(h(x)) = x. This confirms that the composition of f with h returns the original input x. Next, we evaluate h(f(x)). Substituting f(x) into h(x), we get h(f(x)) = h((1/4)x - 12) = 4((1/4)x - 12) + 48. Distributing the 4, we have h(f(x)) = x - 48 + 48, which simplifies to h(f(x)) = x. This also confirms that the composition of h with f returns the original input x. Since both f(h(x)) and h(f(x)) equal x, we have confidently verified that h(x) = 4x + 48 is indeed the inverse function of f(x) = (1/4)x - 12. Verifying the inverse function through composition is an essential practice, ensuring that our calculations are accurate and our understanding of inverse functions is sound.

The Correct Answer: Option D - h(x) = 4x + 48

Having meticulously derived and verified the inverse of the function f(x) = (1/4)x - 12, we arrive at the definitive answer: h(x) = 4x + 48. This corresponds to option D in the given choices. The step-by-step process of finding the inverse, involving replacing f(x) with y, interchanging x and y, and solving for y, has led us to this accurate result. Moreover, the crucial verification step, where we composed the original function with its inverse and confirmed that it returned the original input, has solidified our confidence in the answer. Options A, B, and C can be eliminated as they do not satisfy the defining property of inverse functions. The correct answer, option D, represents the function that precisely reverses the operations of f(x) = (1/4)x - 12, highlighting the elegance and precision of inverse function concepts.

In conclusion, understanding inverse functions is crucial for mastering various mathematical concepts and problem-solving techniques. By carefully following the steps of finding the inverse and verifying the result, we can confidently navigate the world of functions and their inverses. The example of f(x) = (1/4)x - 12 serves as a clear illustration of the process and underscores the importance of accuracy and verification in mathematical endeavors. The journey into inverse functions opens doors to deeper mathematical insights and empowers us to tackle a wider range of problems with greater proficiency.