Solving The Functional Equation F(x/y) = F(x) - F(y) A Detailed Solution

In the fascinating world of functional equations, we often encounter relationships that dictate how a function behaves under certain operations. One such intriguing equation is f(x/y) = f(x) - f(y), where f is a function mapping positive real numbers to real numbers. This equation reveals a hidden structure within the function, linking its values at different points through a simple division and subtraction. The beauty of this equation lies in its ability to connect seemingly disparate values of the function, creating a web of interconnectedness. To solve this equation is to unravel a piece of this mathematical puzzle. This exploration is not merely an academic exercise; functional equations like this arise in various branches of mathematics and its applications. They can model phenomena where relationships between quantities are expressed through functional dependencies, such as in physics, engineering, and economics. Understanding how to solve such equations can lead to deeper insights into the underlying structures of these phenomena.

This article delves into the intricacies of this functional equation, focusing on finding differentiable solutions. We will embark on a journey to understand the properties that a function must possess to satisfy this equation, paying close attention to the role of differentiability. The condition that f is differentiable adds another layer of structure, allowing us to leverage the power of calculus to uncover the function's form. Furthermore, we introduce a limit condition, lim{x → 0} [f(1+x)/x] = 3_, which acts as a crucial piece of the puzzle, guiding us towards a specific solution. This limit provides a local behavior of the function near 1, a critical point that dictates the function's overall structure. The interplay between the functional equation, differentiability, and the limit condition forms the core of our investigation. We will dissect each element, piece together the clues, and ultimately reveal the function that satisfies all the given conditions.

Our initial focus is to dissect the given functional equation: f(x/y) = f(x) - f(y). This equation is the cornerstone of our investigation, and a thorough understanding of its implications is crucial. It essentially states that the function value at the ratio of two numbers is equal to the difference of the function values at those individual numbers. This deceptively simple relationship holds a wealth of information about the function's behavior. This equation belongs to a class of functional equations known as Cauchy's functional equations, which have a rich history and are fundamental in the study of functional analysis. Cauchy's functional equations, in their various forms, appear in many contexts, from the characterization of linear functions to the study of homomorphisms in abstract algebra. The specific form we are dealing with is a variant related to the logarithmic function, which, as we will see, plays a key role in the solution.

To begin our exploration, let's consider some simple substitutions. Setting x = y in the equation, we get f(1) = f(x) - f(x) = 0. This is a significant result, as it reveals that the function f takes the value 0 at x = 1. This point serves as an anchor, a fixed value that will help us navigate the function's behavior across its domain. Next, let's set x = 1. This gives us f(1/y) = f(1) - f(y) = -f(y), since we know f(1) = 0. This result shows that the function exhibits a kind of symmetry: the value at the reciprocal of a number is the negative of the value at the original number. This property is reminiscent of the behavior of logarithmic functions, which we will explore further. Another insightful substitution is setting y = 1/z. This transforms the original equation into f(xz) = f(x) - f(1/z) = f(x) + f(z), using the property we just derived. This transformed equation, f(xz) = f(x) + f(z), is a classic form known as Cauchy's logarithmic functional equation. It's a powerful result, indicating that the function transforms products into sums, a hallmark characteristic of logarithms. This connection to logarithms is a crucial hint, suggesting that our solution might be related to the logarithmic function. The logarithmic functional equation is a cornerstone in the study of logarithms, providing an axiomatic foundation for their properties. Its solutions are deeply connected to the concept of logarithms in various mathematical contexts.

Now, let's incorporate the additional information that f(x) is differentiable on the interval (0, ∞). This condition is a game-changer, allowing us to leverage the tools of calculus to analyze the function's behavior. Differentiability implies that the function is smooth and has a well-defined derivative at every point in its domain. This smoothness allows us to consider the rate of change of the function, providing insights into its local behavior and overall structure. The derivative, f'(x), represents the instantaneous rate of change of f(x) with respect to x, and its existence gives us a powerful handle on the function's properties. The interplay between differentiability and functional equations is a rich area of mathematical investigation, often leading to elegant solutions and deeper understanding of the functions involved.

Along with differentiability, we are given the limit condition: lim_{x → 0} [f(1+x)/x] = 3. This limit provides crucial information about the function's behavior near x = 1. It tells us that as x approaches 0, the ratio of f(1+x) to x approaches 3. This is essentially giving us the derivative of f at x = 1, since the limit resembles the definition of a derivative. This limit condition is a key piece of the puzzle, acting as a boundary condition that helps us pin down the specific solution we are seeking. Limit conditions of this kind are frequently used in the study of differential equations and functional equations, providing crucial information about the function's behavior at specific points.

To effectively use the differentiability condition, we can differentiate the equation f(xz) = f(x) + f(z) with respect to x. This differentiation allows us to bring the derivative f'(x) into the equation, connecting it to the functional relationship. Applying the chain rule, we get z f'(xz) = f'(x). This equation links the derivative at xz to the derivative at x, providing a powerful tool for analysis. Now, let's set x = 1 in this equation. This yields z f'(z) = f'(1). This is a significant result, as it expresses f'(z) in terms of a constant, namely f'(1). We can rewrite this as f'(z) = f'(1)/z. This equation tells us that the derivative of f(z) is inversely proportional to z, a characteristic of logarithmic functions. This reinforces our earlier hint that the solution is related to logarithms. The constant of proportionality, f'(1), is the key to unlocking the specific solution that satisfies our limit condition.

We have now established that the derivative of f(x) is given by f'(x) = f'(1)/x. To find f(x) itself, we need to integrate this expression. Integration is the inverse operation of differentiation, allowing us to reconstruct the function from its derivative. Integrating both sides of the equation with respect to x, we get f(x) = f'(1) ln(x) + C, where C is the constant of integration. This equation reveals that f(x) is indeed a logarithmic function, as we suspected. The natural logarithm, ln(x), is the integral of 1/x, and the constant f'(1) scales the logarithm. The constant of integration, C, represents a vertical shift of the function. To fully determine f(x), we need to find the values of f'(1) and C.

Recall that we found earlier that f(1) = 0. This gives us our first constraint. Substituting x = 1 into the equation f(x) = f'(1) ln(x) + C, we get 0 = f'(1) ln(1) + C. Since ln(1) = 0, this simplifies to C = 0. This eliminates the constant of integration, simplifying our expression to f(x) = f'(1) ln(x). Now, we turn to the limit condition: lim_{x → 0} [f(1+x)/x] = 3. This condition will help us determine the value of f'(1). Substituting f(x) = f'(1) ln(x) into the limit, we get lim_{x → 0} [f'(1) ln(1+x)/x] = 3. We can pull the constant f'(1) out of the limit, giving us f'(1) lim_{x → 0} [ln(1+x)/x] = 3. The limit lim_{x → 0} [ln(1+x)/x] is a well-known limit that equals 1. This limit is a fundamental result in calculus, often used in the derivation of the derivative of the natural logarithm. Therefore, we have f'(1) * 1 = 3, which implies f'(1) = 3. With f'(1) = 3, we can finally write the complete solution for f(x). Substituting this value back into our expression, we get f(x) = 3 ln(x). This is the unique differentiable solution that satisfies both the functional equation and the limit condition. This function beautifully combines the logarithmic nature dictated by the functional equation with the scaling factor provided by the limit condition.

To ensure the validity of our solution, let's verify that f(x) = 3 ln(x) indeed satisfies the given conditions. First, we check the functional equation f(x/y) = f(x) - f(y). Substituting our solution, we get 3 ln(x/y) = 3 ln(x) - 3 ln(y). Using the logarithmic property ln(a/b) = ln(a) - ln(b), we see that this equation holds true. Next, we verify the limit condition: lim_{x → 0} [f(1+x)/x] = 3. Substituting our solution, we get lim_{x → 0} [3 ln(1+x)/x] = 3 lim_{x → 0} [ln(1+x)/x]. As we mentioned earlier, the limit lim_{x → 0} [ln(1+x)/x] equals 1, so the entire expression equals 3, satisfying the limit condition. Finally, we note that f(x) = 3 ln(x) is differentiable on (0, ∞), as the natural logarithm is a smooth function on this interval.

In conclusion, we have successfully determined the function f(x) that satisfies the functional equation f(x/y) = f(x) - f(y), is differentiable on (0, ∞), and meets the limit condition lim_{x → 0} [f(1+x)/x] = 3. Through a combination of algebraic manipulation, differentiation, integration, and careful consideration of limits, we have arrived at the unique solution: f(x) = 3 ln(x). This journey through the world of functional equations has highlighted the power of mathematical tools in unraveling the hidden structures within functions. The interplay between algebraic equations, calculus, and limit conditions has allowed us to pinpoint a specific solution from a vast landscape of possibilities. The function f(x) = 3 ln(x) stands as a testament to the beauty and elegance of mathematical problem-solving, showcasing how seemingly simple equations can lead to profound insights and concrete solutions.