Unraveling A Math Function: Analysis And Primitive Calculation

Hey math enthusiasts! Today, we're going to break down a cool problem involving a numerical function. We'll explore its properties, deconstruct its form, and ultimately find its primitive. It's going to be a fun ride, so buckle up! This guide will provide a comprehensive understanding of the function, ensuring you grasp the core concepts and techniques involved in solving this type of problem. We will use various methods to break down this problem. Let's get started!

Unveiling the Function: Defining f(x)

Let's start by defining our star function. We are given a function f that's defined on the interval I = ]1; +∞[. The function's expression is: f(x) = (2x + 5) / (x - 1)³. Okay, so what does this actually mean? Basically, this formula tells us how to calculate the output of the function (f(x)) for any input value x that's greater than 1. The interval I = ]1; +∞[ means that x can take any value that's strictly greater than 1 (but not including 1 itself) and extending all the way to infinity. This is a crucial detail, as the behavior of the function depends on this domain. Understanding the domain is fundamental; it dictates the possible input values we can feed into our function, which in turn affects the function's output and overall behavior. Without a clear understanding of the domain, any analysis or calculation could lead to incorrect conclusions.

So, with f(x) = (2x + 5) / (x - 1)³, the function's expression tells us that we're dealing with a rational function. The numerator is a linear expression (2x + 5), and the denominator is a cubic expression involving a shift (x - 1)³. This structure immediately gives us some clues about the function's behavior. We can anticipate that it might have a vertical asymptote at x = 1 (because the denominator becomes zero there). Also, as x gets very large, the function may tend towards zero because the cubic term in the denominator will grow much faster than the linear term in the numerator. f(x) = (2x + 5) / (x - 1)³ is not just any function; it's a specific mathematical entity with its own unique properties and characteristics. The subsequent steps are all about exploring these properties. This will help you to further comprehend the inner workings of the function. Now we know the basics, let's explore it more.

Decomposing the Function: Finding a and b

Here’s where things get interesting! The first part of our mission is to determine two real numbers, a and b, such that we can rewrite f(x) in a different, more convenient form. We want to find a and b so that:

f(x) = a / (x - 1)² + b / (x - 1)³.

This is a classic technique called partial fraction decomposition. The goal is to break down a complex fraction into simpler fractions. Why do we do this? Because it makes other operations, like finding the integral (the primitive function in our case), much easier. To find a and b, we're essentially trying to reverse the process of adding fractions. The initial fraction might seem complicated, but breaking it down into a sum of simpler fractions simplifies our approach to analysis and integration. By strategically choosing the denominators, we hope to make the decomposition straightforward. This approach greatly simplifies the process of finding the primitive function. The process involves some algebraic manipulation.

To find the values of a and b, we first need to combine the fractions on the right-hand side and equate the resulting expression to the original f(x). We can rewrite the right-hand side with a common denominator of (x - 1)³: a / (x - 1)² + b / (x - 1)³ = [a(x - 1) + b] / (x - 1)³. Now we can set up an equation. We know that f(x) = (2x + 5) / (x - 1)³. So, we can set up the equation (2x + 5) / (x - 1)³ = [a(x - 1) + b] / (x - 1)³. Since the denominators are the same, we can equate the numerators: 2x + 5 = a(x - 1) + b. This is the core of our strategy. The process of partial fraction decomposition hinges on the ability to equate the coefficients of the powers of x. This allows us to create a system of equations, which we can solve to find the unknown values of a and b. The process of finding these numbers is both systematic and strategic. Therefore, we can find the value of a and b. Next, we equate the coefficients. By expanding the right-hand side, we get 2x + 5 = ax - a + b. Now we can equate the coefficients of the x terms and the constant terms on both sides. This gives us two equations: a = 2 (from the x terms) and -a + b = 5 (from the constant terms). We can see that the coefficient of x on the left side is 2, while on the right side, it's a. Thus, a = 2. With a = 2, we can substitute this value into the second equation: -2 + b = 5. Solving for b, we find that b = 7. So, we have determined that a = 2 and b = 7. Let's celebrate our findings.

Now we know that f(x) = 2 / (x - 1)² + 7 / (x - 1)³. This decomposition is really useful because it separates the function into simpler terms that are easier to integrate. Now that we've found our a and b values, we're one step closer to solving our original problem. It's time to find the primitive function.

Finding the Primitive: Unveiling F(x)

Okay, time for the grand finale: finding the primitive function F of f. The primitive function is essentially the antiderivative of f. In other words, if you take the derivative of F(x), you should get f(x). From the last part, we have f(x) = 2 / (x - 1)² + 7 / (x - 1)³. Finding the primitive of f means finding a function F(x) whose derivative is f(x). We can integrate each term separately. The power rule is super handy here. The process of finding the primitive often involves the reverse application of differentiation rules. The goal is to determine the function whose derivative matches the given expression. This process is the key to solving our problem.

We know that the integral of 1 / (x - 1)² is -1 / (x - 1) (plus a constant). And the integral of 1 / (x - 1)³ is -1 / [2(x - 1)²] (plus a constant). Now we need to find the primitive. Therefore, F(x) will be of the form: F(x) = ∫ f(x) dx = ∫ [2 / (x - 1)² + 7 / (x - 1)³] dx. Integrating each term, we get: F(x) = -2 / (x - 1) - 7 / [2(x - 1)²] + C, where C is the constant of integration. We're not quite done yet, because the problem states that the primitive function F must vanish at x = 2. This means that F(2) = 0. So we need to figure out the value of C. It's important to remember that when finding the primitive of a function, we must include an arbitrary constant of integration (C). This is because the derivative of any constant is zero, which means that any constant could be added to the primitive function without affecting its derivative. Now, let’s find the value of C. We have F(2) = -2 / (2 - 1) - 7 / [2(2 - 1)²] + C = 0. Simplifying, we get -2 - 7/2 + C = 0. Thus, -11/2 + C = 0, which means C = 11/2. So, the final expression of our primitive function is: F(x) = -2 / (x - 1) - 7 / [2(x - 1)²] + 11/2. And there you have it! We've successfully determined the primitive function of f that meets the specified conditions. We have now fully solved the problem.

Summary and Key Takeaways

Here's what we've achieved: We started with a function, f(x) = (2x + 5) / (x - 1)³, and we wanted to find its primitive F(x). We used partial fraction decomposition to rewrite f(x) as f(x) = 2 / (x - 1)² + 7 / (x - 1)³. We then integrated this to find F(x) = -2 / (x - 1) - 7 / [2(x - 1)²] + C, and finally, we used the condition F(2) = 0 to find the constant of integration, resulting in the final answer: F(x) = -2 / (x - 1) - 7 / [2(x - 1)²] + 11/2. The problem perfectly illustrates the relationship between a function and its primitive function. We have used decomposition to find the primitive function. This shows how different mathematical techniques can be combined to solve complex problems.

This entire exercise emphasized the importance of mastering fundamental concepts like function decomposition, integration, and the concept of a constant of integration. By working through the problem step-by-step, we've demonstrated how to analyze a function, manipulate its form for easier integration, and finally find its primitive function that satisfies specific conditions. We hope this guide has been helpful! Remember, the more you practice, the better you'll get at these types of problems. Keep up the amazing work, and keep exploring the wonderful world of mathematics! Keep experimenting and enjoy the process of learning.