Finding (f/g)(x): A Step-by-Step Guide

Have you ever wondered how to combine two functions through division? This guide will walk you through the process of finding (f/g)(x), where f(x) and g(x) are given functions. Specifically, we'll tackle the example where f(x) = -x² + 5x and g(x) = -x + 4. This might seem daunting at first, but by breaking it down into manageable steps, you'll see it's quite straightforward. We'll explore the concept of function division, demonstrate the substitution process, and simplify the resulting expression. By the end of this guide, you'll be equipped to solve similar problems and have a solid understanding of function operations. So, let's dive in and unravel the mystery of (f/g)(x)!

Understanding Function Division

Before we jump into the specific problem, let's solidify our understanding of function division. At its core, finding (f/g)(x) means dividing the function f(x) by the function g(x). It's essential to remember that this operation is only valid when g(x) ≠ 0, as division by zero is undefined in mathematics. This condition introduces a crucial consideration: we need to identify any values of x that would make g(x) equal to zero and exclude them from the domain of (f/g)(x). The concept of function division is a fundamental operation in calculus and algebra, enabling us to create new functions by relating existing ones. Understanding function division opens the door to exploring more complex mathematical models and relationships. It allows us to analyze how two functions interact and how their combined behavior manifests. For instance, in real-world applications, function division might represent the ratio of two quantities, such as the ratio of profit to revenue or the ratio of population growth to available resources. Grasping the nuances of function division empowers us to tackle a broader range of mathematical challenges and apply these principles to practical scenarios.

Setting up (f/g)(x)

In our case, we are given f(x) = -x² + 5x and g(x) = -x + 4. So, finding (f/g)(x) simply means substituting these expressions into the fraction f(x)/g(x). This gives us (f/g)(x) = (-x² + 5x) / (-x + 4). This is the initial setup, but it's not the final answer. We need to simplify this expression further. Think of this step as laying the foundation for the solution. We've taken the given functions and expressed them in the form we need for division. The next step involves looking for opportunities to simplify the fraction. Simplification might involve factoring, canceling common factors, or performing polynomial division. The key is to manipulate the expression in a way that makes it easier to understand and work with. This initial setup is crucial because it sets the stage for the subsequent steps. Without a clear setup, the simplification process can become much more challenging. So, take your time with this step and ensure you've accurately substituted the given functions into the correct positions. This careful approach will pave the way for a smooth and successful solution.

Simplifying the Expression

Now comes the crucial step of simplifying the rational function. Looking at the numerator, -x² + 5x, we can see that it has a common factor of -x. Factoring out -x, we get -x(x - 5). So, our expression now looks like: (f/g)(x) = -x(x - 5) / (-x + 4). At this point, we need to assess whether further simplification is possible. We look for common factors between the numerator and the denominator that can be canceled out. In this case, there are no immediately obvious common factors. The expression (x - 5) in the numerator is not directly compatible with the expression (-x + 4) in the denominator. This suggests that we cannot simplify the expression by direct cancellation. However, this doesn't mean we're finished. It simply means we've exhausted the most straightforward simplification techniques. The expression we have now, -x(x - 5) / (-x + 4), is indeed the simplified form of the rational function. We've taken the initial expression and manipulated it to its most basic representation. This simplified form is valuable because it allows us to analyze the function's behavior more easily. We can now readily identify the function's zeros, its asymptotes, and its overall trend. Simplification is a core skill in mathematics, allowing us to transform complex expressions into more manageable forms. It's a process that requires careful observation, strategic manipulation, and a keen eye for potential simplifications.

The Final Answer

Therefore, the simplified form of (f/g)(x) for the given functions f(x) = -x² + 5x and g(x) = -x + 4 is (f/g)(x) = -x(x - 5) / (-x + 4). This is the final answer, expressed as a rational function in its simplest form. We have successfully navigated the process of function division, from the initial setup to the final simplification. It's important to recognize that this answer represents a single function that results from the division of the two original functions. This new function inherits properties from both f(x) and g(x), but it also possesses its own unique characteristics. The denominator, (-x + 4), plays a crucial role in defining the function's domain. We know that the denominator cannot be equal to zero, so we must exclude any values of x that would make (-x + 4) equal to zero. In this case, x = 4 is the value that needs to be excluded. This exclusion is vital for understanding the function's behavior and avoiding mathematical errors. The final answer, (f/g)(x) = -x(x - 5) / (-x + 4), is not just a mathematical expression; it's a representation of a relationship between two functions. It encapsulates the essence of function division and provides a foundation for further analysis and application.

In conclusion, determining (f/g)(x) involves dividing the function f(x) by g(x) and simplifying the resulting expression. For f(x) = -x² + 5x and g(x) = -x + 4, we found that (f/g)(x) = -x(x - 5) / (-x + 4). Remember to always consider the domain of the resulting function, ensuring that the denominator is not zero. This process demonstrates a fundamental operation in mathematics, applicable in various fields. For further exploration of rational functions and their applications, you can visit resources like Khan Academy.