Evaluating (f/g)(5) With F(x)=7+4x And G(x)=1/(2x) A Step-by-Step Guide

by ADMIN 72 views

In the realm of mathematics, understanding how to operate on functions is crucial. Functions are the building blocks of many mathematical models, and learning how to combine them through operations like addition, subtraction, multiplication, and division allows us to create more complex and nuanced models. In this article, we delve into the concept of function division and explore how to evaluate the result at a specific point. Specifically, we will tackle the problem of finding the value of (f/g)(5), given that f(x) = 7 + 4x and g(x) = 1/(2x). This problem serves as an excellent example of how to combine algebraic manipulation with function evaluation to arrive at a solution.

Keywords: function operations, function division, algebraic manipulation, function evaluation, f(x) = 7 + 4x, g(x) = 1/(2x), (f/g)(5). This article provides a comprehensive guide to solving this type of problem, focusing on clear explanations and step-by-step solutions. Whether you are a student learning about function operations or simply someone who enjoys mathematical challenges, this article offers valuable insights and practical techniques. By the end of this exploration, you will be well-equipped to handle similar problems involving function division and evaluation.

Function division, denoted as (f/g)(x), is a fundamental operation in mathematics that combines two functions, f(x) and g(x), by dividing the first function by the second. The result is a new function that represents the ratio of the two original functions. However, it's important to remember that function division comes with a critical caveat: we must ensure that the denominator, g(x), is not equal to zero. Division by zero is undefined in mathematics, and we must always be mindful of this constraint when working with function division.

The formal definition of function division is straightforward: (f/g)(x) = f(x) / g(x), provided that g(x) ≠ 0. This means that to find the value of (f/g)(x) for a specific x, we first evaluate f(x) and g(x) separately, and then divide the result of f(x) by the result of g(x). However, we must always check that the value of g(x) is not zero for the given x. If g(x) = 0, then (f/g)(x) is undefined at that point. Understanding this restriction is crucial for avoiding errors and ensuring the validity of our calculations. This article provides a detailed exploration of function division, including the importance of considering the domain of the resulting function. By understanding the definition and limitations of function division, we can confidently apply this operation to solve a variety of mathematical problems.

Let's break down the process of finding (f/g)(5) step-by-step. We are given two functions: f(x) = 7 + 4x and g(x) = 1/(2x). Our goal is to find the value of the function (f/g)(x) when x = 5. This involves a combination of algebraic manipulation and function evaluation, which we will tackle systematically.

Keywords: step-by-step solution, evaluating (f/g)(5), f(x) = 7 + 4x, g(x) = 1/(2x), algebraic manipulation, function evaluation. First, we need to find the expression for (f/g)(x). According to the definition of function division, (f/g)(x) = f(x) / g(x). Substituting the given functions, we get (f/g)(x) = (7 + 4x) / (1/(2x)). This expression involves a fraction in the denominator, which can be simplified by multiplying the numerator by the reciprocal of the denominator. In this case, the reciprocal of 1/(2x) is 2x. Therefore, (f/g)(x) = (7 + 4x) * (2x). Now, we can simplify this expression by distributing the 2x across the terms in the parentheses: (f/g)(x) = 2x * 7 + 2x * 4x = 14x + 8x². This gives us a simplified expression for (f/g)(x). Next, we need to evaluate this expression at x = 5. To do this, we substitute 5 for x in the expression (f/g)(x) = 14x + 8x². This gives us (f/g)(5) = 14 * 5 + 8 * 5². Calculating this, we get (f/g)(5) = 70 + 8 * 25 = 70 + 200 = 270. Thus, the value of (f/g)(5) is 270. This step-by-step solution provides a clear path for evaluating function division at a specific point, combining algebraic manipulation with careful substitution and calculation.

  1. Define Function Division: Recall that (f/g)(x) = f(x) / g(x).
  2. Substitute Functions: Substitute the given functions f(x) and g(x) into the expression. (f/g)(x) = (7 + 4x) / (1/(2x))
  3. Simplify the Expression: To simplify, multiply the numerator by the reciprocal of the denominator. (f/g)(x) = (7 + 4x) * (2x) (f/g)(x) = 14x + 8x²
  4. Evaluate at x = 5: Substitute x = 5 into the simplified expression. (f/g)(5) = 14 * 5 + 8 * 5² (f/g)(5) = 70 + 8 * 25 (f/g)(5) = 70 + 200 (f/g)(5) = 270

While the step-by-step solution provides a direct method for evaluating (f/g)(5), exploring alternate approaches can deepen our understanding of function operations and provide valuable problem-solving skills. One such approach involves evaluating f(5) and g(5) separately before performing the division. This method highlights the individual contributions of each function and can sometimes simplify the calculations.

Keywords: alternate approaches, different perspectives, evaluating f(5), evaluating g(5), simplifying calculations, function operations. First, let's evaluate f(5). Given f(x) = 7 + 4x, we substitute x = 5 to get f(5) = 7 + 4 * 5 = 7 + 20 = 27. Next, we evaluate g(5). Given g(x) = 1/(2x), we substitute x = 5 to get g(5) = 1/(2 * 5) = 1/10. Now that we have f(5) and g(5), we can find (f/g)(5) by dividing f(5) by g(5). This gives us (f/g)(5) = f(5) / g(5) = 27 / (1/10). Dividing by a fraction is the same as multiplying by its reciprocal, so (f/g)(5) = 27 * 10 = 270. This result matches the answer we obtained using the step-by-step method, which reinforces the correctness of our solution. This alternate approach demonstrates the flexibility of function operations and the potential for multiple paths to the same answer. By considering different perspectives, we can develop a more robust understanding of mathematical concepts and improve our problem-solving abilities. This section provides a valuable comparison of methods, showcasing the efficiency and accuracy of both approaches.

  1. Evaluate f(5): Substitute x = 5 into f(x) = 7 + 4x. f(5) = 7 + 4 * 5 = 27
  2. Evaluate g(5): Substitute x = 5 into g(x) = 1/(2x). g(5) = 1/(2 * 5) = 1/10
  3. Divide f(5) by g(5): (f/g)(5) = f(5) / g(5). (f/g)(5) = 27 / (1/10) (f/g)(5) = 27 * 10 = 270

When working with function operations, particularly function division, it's crucial to be aware of common mistakes that can lead to incorrect answers. One of the most frequent errors is failing to simplify the expression for (f/g)(x) before evaluating it at a specific point. This can result in more complex calculations and increase the likelihood of making a mistake. Another common mistake is forgetting to consider the domain of the resulting function. As mentioned earlier, we must ensure that the denominator, g(x), is not equal to zero. If g(x) = 0 for a particular x-value, then (f/g)(x) is undefined at that point.

Keywords: common mistakes, avoiding errors, simplifying expressions, domain of function, division by zero, function operations. Additionally, errors can occur during the algebraic manipulation process, such as incorrectly distributing terms or making mistakes when multiplying or dividing fractions. Careful attention to detail and a systematic approach can help minimize these errors. It's also important to double-check your work, especially when dealing with multiple steps or complex expressions. By being aware of these common mistakes and taking steps to avoid them, you can ensure the accuracy of your calculations and gain confidence in your ability to work with function operations. This section provides practical guidance on error prevention, emphasizing the importance of thoroughness and attention to detail in mathematical problem-solving.

While function operations may seem like an abstract mathematical concept, they have numerous real-world applications across various fields. Understanding how to combine functions through operations like division can help us model and analyze complex phenomena in areas such as physics, engineering, economics, and computer science. In physics, for example, the concept of average velocity can be represented as the division of the displacement function by the time function. Similarly, in economics, cost-benefit analysis often involves comparing two functions – the cost function and the benefit function – using division to determine the efficiency of a particular project or investment.

Keywords: real-world applications, physics, engineering, economics, computer science, cost-benefit analysis, modeling complex phenomena. In engineering, function division can be used to analyze the performance of systems, such as calculating the gain of an amplifier by dividing the output signal function by the input signal function. In computer science, function operations are fundamental to algorithm design and analysis, where functions are used to represent computational processes. Understanding these real-world applications can provide a deeper appreciation for the practical significance of function operations and motivate further exploration of mathematical concepts. This section highlights the relevance of mathematics to everyday life, demonstrating how seemingly abstract concepts can be used to solve real-world problems.

In this article, we've explored the concept of function division and demonstrated how to evaluate the result at a specific point. We tackled the problem of finding (f/g)(5), given f(x) = 7 + 4x and g(x) = 1/(2x), using both a step-by-step approach and an alternate method. We also discussed common mistakes to avoid and highlighted the real-world applications of function operations. By understanding the definition of function division, practicing algebraic manipulation, and paying attention to detail, you can master this essential mathematical skill. Function operations are a cornerstone of mathematical modeling and analysis, and a solid understanding of these concepts will serve you well in various academic and professional pursuits. This article provides a comprehensive overview of function division, equipping readers with the knowledge and skills necessary to confidently tackle related problems. We encourage you to continue exploring function operations and their applications to further enhance your mathematical understanding.