Finding And Simplifying (f+g)(x) Where F(x)=7x+3 And G(x)=4x-5
Introduction
In mathematics, especially in the study of functions, combining functions through various operations is a fundamental concept. One such operation is the addition of functions. When we add two functions, say f(x) and g(x), we create a new function, denoted as (f+g)(x). This new function is simply the sum of the individual functions at each point in their domain. In this article, we will delve into the process of finding and simplifying the function (f+g)(x) given two specific functions: f(x) = 7x + 3 and g(x) = 4x - 5. Understanding how to perform this operation is crucial for various mathematical applications, including calculus, algebra, and real-world modeling.
Our primary objective here is to provide a clear and step-by-step guide on how to add these two functions and simplify the resulting expression. We will begin by defining the concept of function addition, then proceed to apply this definition to the given functions. By combining like terms, we will arrive at the simplified form of (f+g)(x). This process not only reinforces the basic principles of algebra but also highlights the importance of careful manipulation of algebraic expressions. Whether you are a student learning about functions for the first time or someone looking to refresh your understanding, this article aims to offer a comprehensive and easy-to-follow explanation.
This article serves as a detailed guide to understand the process of function addition, specifically focusing on finding and simplifying (f+g)(x). Through clear explanations and step-by-step calculations, we aim to demystify the process and equip you with the necessary skills to tackle similar problems. Whether you're a student, educator, or simply someone with an interest in mathematics, this article provides valuable insights into the world of function operations. So, let's embark on this mathematical journey and explore the fascinating realm of function addition!
Understanding Function Addition
Function addition is a fundamental operation in mathematics that combines two or more functions to create a new function. Given two functions, f(x) and g(x), their sum, denoted as (f+g)(x), is defined as the sum of the functions' values at each point x in their common domain. Mathematically, this can be expressed as (f+g)(x) = f(x) + g(x). This operation is straightforward but essential for various applications in calculus, algebra, and other branches of mathematics. The concept of function addition is not just a theoretical exercise; it has practical implications in modeling real-world phenomena where different functions might represent various aspects of a system.
To illustrate this concept, consider two functions representing the cost and revenue of a business. By adding these functions, we can obtain a new function that represents the total profit or loss. Similarly, in physics, adding functions can represent the combined effect of multiple forces or the total energy of a system. The key to understanding function addition lies in recognizing that we are simply adding the outputs of the functions for a given input. This process is analogous to adding numbers, but instead of individual values, we are adding expressions that define the functions.
Before we delve into the specific example of f(x) = 7x + 3 and g(x) = 4x - 5, it is crucial to grasp the basic principles of function addition. The resulting function (f+g)(x) is a new function that inherits its domain from the common domain of f(x) and g(x). This means that the values of x for which (f+g)(x) is defined are only those for which both f(x) and g(x) are defined. In most cases, especially when dealing with polynomial functions, the domain is all real numbers. However, when dealing with functions that have restrictions on their domains, such as rational functions or square root functions, it is essential to consider these restrictions when determining the domain of (f+g)(x). In summary, function addition is a powerful tool for combining functions, with broad applications in various fields. By understanding the basic principles and domain considerations, we can effectively use this operation to solve complex problems.
Step-by-Step Solution
To find and simplify the function (f+g)(x) where f(x) = 7x + 3 and g(x) = 4x - 5, we will follow a step-by-step approach. This methodical process will ensure clarity and accuracy in our calculations. The first step involves applying the definition of function addition, which states that (f+g)(x) = f(x) + g(x). By substituting the given functions into this definition, we set the stage for combining like terms and simplifying the expression. This initial step is crucial as it lays the foundation for the subsequent algebraic manipulations.
Next, we substitute the expressions for f(x) and g(x) into the equation. This yields (f+g)(x) = (7x + 3) + (4x - 5). At this point, it's essential to pay close attention to the signs and parentheses to avoid any errors. The goal is to combine the terms correctly, and this requires a careful examination of the expression. Once the substitution is complete, we proceed to the next step, which involves combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, the like terms are the terms involving x (i.e., 7x and 4x) and the constant terms (i.e., 3 and -5).
Combining like terms involves adding or subtracting the coefficients of the terms with the same variable and adding or subtracting the constant terms. In our expression, we add 7x and 4x to get 11x, and we add 3 and -5 to get -2. This process simplifies the expression significantly. After combining like terms, we arrive at the simplified form of the function (f+g)(x). The resulting expression should be as concise as possible, with no further terms that can be combined. In this case, the simplified function is (f+g)(x) = 11x - 2. This final expression represents the sum of the two original functions, and it is the answer we were seeking. By following these steps, we have successfully found and simplified the function (f+g)(x).
Detailed Calculation
Let's walk through the detailed calculation to find (f+g)(x) given f(x) = 7x + 3 and g(x) = 4x - 5. The first step, as previously mentioned, is to apply the definition of function addition: (f+g)(x) = f(x) + g(x). This definition provides the framework for our calculation and ensures that we are on the right track. Understanding this basic principle is crucial for solving similar problems in the future. Once we have established this foundation, we can proceed to the next step, which involves substituting the given expressions for f(x) and g(x).
Substituting f(x) = 7x + 3 and g(x) = 4x - 5 into the equation (f+g)(x) = f(x) + g(x), we get (f+g)(x) = (7x + 3) + (4x - 5). This substitution is a direct application of the definition and sets the stage for further simplification. It is important to note that the parentheses are used to maintain the correct order of operations and to ensure that the signs are handled correctly. Once the substitution is complete, the next step is to combine like terms. This involves identifying and grouping terms with the same variable and constant terms.
To combine like terms, we first identify the terms with the variable x, which are 7x and 4x. Adding these terms together, we get 7x + 4x = 11x. Next, we identify the constant terms, which are 3 and -5. Adding these terms together, we get 3 + (-5) = -2. Now, we combine these results to form the simplified expression for (f+g)(x). This gives us (f+g)(x) = 11x - 2. This final expression is the simplified form of the function addition, and it represents the sum of the two original functions. The detailed calculation demonstrates how each step contributes to the final result, providing a clear and comprehensive understanding of the process.
Conclusion
In conclusion, we have successfully found and simplified the function (f+g)(x), where f(x) = 7x + 3 and g(x) = 4x - 5. By following a step-by-step approach, we first applied the definition of function addition, which states that (f+g)(x) = f(x) + g(x). Then, we substituted the given expressions for f(x) and g(x) into this equation, resulting in (f+g)(x) = (7x + 3) + (4x - 5). Finally, we combined like terms to simplify the expression, leading to the solution (f+g)(x) = 11x - 2. This process highlights the importance of understanding fundamental mathematical principles and applying them systematically to solve problems.
The ability to add functions is a crucial skill in various areas of mathematics and its applications. It allows us to combine different mathematical models to create more complex and realistic representations of real-world phenomena. For example, in economics, we might add cost and revenue functions to determine profit. In physics, we might add force vectors to determine the net force acting on an object. The applications are vast and varied, making function addition a fundamental tool in mathematical analysis.
Throughout this article, we have emphasized the importance of clarity and accuracy in mathematical calculations. Each step in the process was explained in detail, ensuring that the reader can follow along and understand the reasoning behind each operation. By breaking down the problem into manageable steps, we have demystified the process of function addition and provided a solid foundation for further exploration of mathematical concepts. The solution (f+g)(x) = 11x - 2 is not just an answer; it is the result of a logical and methodical approach to problem-solving. This approach can be applied to a wide range of mathematical problems, making it a valuable skill for students and professionals alike.
