Solving (f - G)(x) = 0 Given F(t) = ISt = X And G(t) = Iφt = 6

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Introduction

In the realm of mathematics, solving equations is a fundamental skill. This article dives deep into finding the value of x for which the equation (f - g)(x) = 0 holds true, given the functions f(t) = iSt = x and g(t) = iφt = 6. This problem intertwines complex numbers and function evaluation, demanding a methodical approach. We will embark on a step-by-step journey, unraveling the complexities and arriving at the solution, ensuring a comprehensive understanding of the underlying principles and techniques involved. Our aim is not just to provide the answer but to elucidate the process, making it accessible and informative for readers of all backgrounds. By breaking down the problem into manageable parts and explaining each step in detail, we hope to empower you with the knowledge and confidence to tackle similar mathematical challenges.

Understanding the Given Functions

Before we dive into solving the equation, it's crucial to have a clear grasp of the functions we're dealing with. Let's break down f(t) = iSt = x and g(t) = iφt = 6. The function f(t) involves the imaginary unit i, a variable S, and t. The product of i, S, and t is equated to x. This suggests that x is a complex number, given the presence of i. The function g(t) also involves the imaginary unit i, a variable φ, and t. In this case, the product iφt is equal to 6. It's important to note that φ is the Greek letter phi, often used to represent an angle or a constant. Understanding the nature of these functions—complex-valued and involving variables and constants—is the first step toward solving the equation. We need to carefully consider how these components interact to determine the value of x that satisfies the given condition. The use of complex numbers adds an extra layer of complexity, requiring us to recall the properties of imaginary units and their manipulation in algebraic expressions.

Setting up the Equation (f - g)(x) = 0

The equation we need to solve is (f - g)(x) = 0. This represents the difference between the functions f and g, evaluated at the value x, being equal to zero. In simpler terms, we're looking for the value of x where the outputs of the two functions are the same. To proceed, we need to express (f - g)(x) in terms of the given functions. This means substituting f(x) and g(x) into the expression. Remember that f(t) = iSt = x and g(t) = iφt = 6. The notation (f - g)(x) indicates that we first evaluate the functions at the value x and then subtract the result of g(x) from the result of f(x). Setting this difference equal to zero gives us an equation that we can then solve for x. This step is crucial because it translates the functional notation into a concrete algebraic equation, which we can manipulate using standard mathematical techniques. The ability to correctly set up the equation is often the most critical part of solving mathematical problems, as it lays the foundation for all subsequent steps.

Substituting and Simplifying

Now, let's substitute the given functions into the equation (f - g)(x) = 0. We know that f(t) = iSt = x, so when we evaluate f at x, we have f(x). However, there's a subtle point here: the function f is defined in terms of t, and its output is equal to x. This means we need to think about what input value t would produce the output x. The expression iSt = x tells us that t = x / (iS). Similarly, for g(t) = iφt = 6, evaluating g at x is not as straightforward because g is a constant function with respect to x as g(t) always equal to 6. Thus we can think of the value of g(x) as the constant 6. The original equation is f(x) - g(x) = 0, and based on the functions definitions f(x) = x and g(x) = 6, substituting gives us x - 6 = 0. This simplified equation is much easier to solve. The key here was recognizing how the function definitions relate to the variable x and how to correctly substitute them into the equation. Simplification is a crucial step in problem-solving, as it reduces complex expressions to their most basic form, making them easier to manipulate and understand.

Solving for x

With the equation simplified to x - 6 = 0, solving for x becomes a straightforward task. To isolate x, we simply need to add 6 to both sides of the equation. This gives us x = 6. This is the value of x that satisfies the original equation (f - g)(x) = 0. It means that when x is equal to 6, the difference between the functions f(x) and g(x) is zero. This solution is a real number, despite the initial presence of the imaginary unit i in the function definitions. This highlights the importance of carrying out the algebraic manipulations carefully to arrive at the correct answer. Solving for a variable is a fundamental skill in algebra, and it often involves a series of steps to isolate the variable on one side of the equation. In this case, the simplicity of the equation allowed us to solve it in just one step, but more complex equations may require multiple steps and different techniques.

Verification of the Solution

To ensure the correctness of our solution, it's always a good practice to verify it. We found that x = 6, so let's plug this value back into the original equation (f - g)(x) = 0. We have f(x) = x, so f(6) = 6. We also have g(x) = 6. Therefore, (f - g)(6) = f(6) - g(6) = 6 - 6 = 0. This confirms that our solution x = 6 is indeed correct. Verification is a critical step in the problem-solving process, as it helps to catch any errors that may have occurred during the solution process. It provides confidence in the answer and ensures that it satisfies the given conditions. By plugging the solution back into the original equation, we can directly check if the equation holds true. This simple step can save time and effort in the long run by preventing incorrect answers from being used in subsequent calculations or applications.

Conclusion

In conclusion, we have successfully found the value of x for which the equation (f - g)(x) = 0 holds true, given the functions f(t) = iSt = x and g(t) = iφt = 6. By carefully analyzing the functions, setting up the equation, substituting the functions, simplifying, solving for x, and verifying the solution, we arrived at the answer x = 6. This problem showcased the importance of understanding function notation, algebraic manipulation, and the properties of complex numbers. The process we followed highlights a systematic approach to solving mathematical problems, emphasizing the need for clarity, precision, and verification. We hope this detailed explanation has provided valuable insights and enhanced your problem-solving skills. Mathematics is a field that rewards careful thinking and methodical approaches, and this example serves as a testament to that principle.