Simplifying (f + G)(x) When F(x) Is 3x + 1 And G(x) Is X - 5

In the realm of mathematics, combining functions is a fundamental operation that allows us to create new functions with unique properties. One common way to combine functions is through addition, denoted as (f + g)(x). This operation involves adding the expressions of two functions, f(x) and g(x), to produce a new function that represents their combined effect. In this comprehensive exploration, we will delve into the concept of (f + g)(x) and meticulously walk through the process of finding its simplest form when given specific functions f(x) and g(x). Specifically, we will consider the case where f(x) = 3x + 1 and g(x) = x - 5, providing a clear and detailed explanation that will benefit students, educators, and anyone interested in mastering this essential mathematical concept. Let's embark on this journey to unravel the intricacies of function addition and simplification.

Defining Function Addition: (f + g)(x)

In mathematical terms, when we talk about adding two functions, f(x) and g(x), we're essentially creating a new function that represents the sum of their individual outputs for any given input x. This new function is denoted as (f + g)(x), and its formal definition is:

(f + g)(x) = f(x) + g(x)

This definition tells us that to find the value of the combined function (f + g)(x) at a particular x, we simply need to add the values of f(x) and g(x) at that same x. It's a straightforward concept, but its applications are vast and varied, appearing in numerous mathematical and real-world contexts. Understanding function addition is crucial for simplifying complex expressions, solving equations, and modeling various phenomena. When dealing with real-world problems, functions often represent different aspects of a situation, and adding them together can provide a holistic view. For example, in economics, f(x) might represent the cost function and g(x) the revenue function; their sum would give the profit function. Similarly, in physics, functions could represent different forces acting on an object, and their sum would yield the net force. Thus, mastering the addition of functions is not just an academic exercise but a practical skill with wide-ranging utility.

The Specific Case: f(x) = 3x + 1 and g(x) = x - 5

Now, let's narrow our focus to the specific functions given in the problem: f(x) = 3x + 1 and g(x) = x - 5. These are both linear functions, which means they represent straight lines when graphed. The function f(x) = 3x + 1 has a slope of 3 and a y-intercept of 1, while g(x) = x - 5 has a slope of 1 and a y-intercept of -5. Linear functions are the simplest type of functions and are widely used in various fields due to their straightforward nature. Understanding how to manipulate and combine linear functions is a foundational skill in algebra and calculus. In real-world applications, linear functions can model relationships between quantities that change at a constant rate, such as the distance traveled by a car moving at a constant speed or the cost of producing a certain number of items. The coefficients and constants in these functions have real-world interpretations; for example, the slope might represent the rate of change, and the y-intercept might represent an initial value. Thus, by working with specific functions like these, we can gain a deeper understanding of how mathematical concepts relate to practical situations. Combining these functions will give us a new linear function, and our goal is to express this combined function in its simplest form.

Step-by-Step Solution to Find (f + g)(x)

To find (f + g)(x), we will follow the definition of function addition: (f + g)(x) = f(x) + g(x). We will substitute the given expressions for f(x) and g(x) and then simplify the resulting expression by combining like terms. This step-by-step approach ensures clarity and reduces the chance of errors, making the process easy to follow and understand. Each step is logically connected to the previous one, building a coherent solution that can be easily replicated for similar problems. The emphasis is on understanding the underlying principles rather than just memorizing the steps. By breaking down the problem into smaller, manageable parts, we can gain a deeper appreciation for the elegance and power of mathematical operations. This methodical approach is not only useful for this specific problem but also for tackling more complex mathematical challenges in the future. Let's begin the process of substituting and simplifying.

Step 1: Substitute f(x) and g(x)

We start by substituting the expressions for f(x) and g(x) into the equation (f + g)(x) = f(x) + g(x). Given that f(x) = 3x + 1 and g(x) = x - 5, we have:

(f + g)(x) = (3x + 1) + (x - 5)

This substitution is the crucial first step in combining the functions. It replaces the abstract notation of f(x) and g(x) with their concrete algebraic expressions. This step is essential because it transforms the problem from a conceptual one to an algebraic one, which can be solved using established rules and techniques. The use of parentheses here is deliberate and important; it ensures that we are adding the entire expression for each function, especially when dealing with functions that involve subtraction or negative signs. Correctly substituting the expressions is paramount because any error at this stage will propagate through the rest of the solution. Therefore, it's always a good practice to double-check the substitution to ensure accuracy before proceeding to the next step. This meticulous attention to detail is a hallmark of effective problem-solving in mathematics.

Step 2: Simplify the Expression

Now that we have (f + g)(x) = (3x + 1) + (x - 5), our next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with 'x' (3x and x) and two constant terms (1 and -5). To simplify, we will group these like terms together and then perform the addition or subtraction.

First, let's remove the parentheses. Since we are adding the expressions, the parentheses do not change the signs of the terms inside them. So, we can rewrite the expression as:

(f + g)(x) = 3x + 1 + x - 5

Next, we group the like terms:

(f + g)(x) = (3x + x) + (1 - 5)

Now, we combine the like terms. 3x + x is equal to 4x, and 1 - 5 is equal to -4. Therefore, the simplified expression is:

(f + g)(x) = 4x - 4

This simplified form is the answer we are looking for. It represents the combined function (f + g)(x) in its most concise form. The process of combining like terms is a fundamental skill in algebra, and it's essential for simplifying expressions and solving equations. By simplifying, we make the expression easier to understand and work with. The result, 4x - 4, is a linear function with a slope of 4 and a y-intercept of -4. This new function represents the combined effect of the original functions f(x) and g(x).

The Answer in Simplest Form

Therefore, the simplest form of (f + g)(x) when f(x) = 3x + 1 and g(x) = x - 5 is:

(f + g)(x) = 4x - 4

This final result is a concise and clear representation of the combined function. It tells us that for any input x, the output of the combined function is 4 times x, minus 4. This simple linear equation can be easily graphed, analyzed, and used for further calculations. The journey from the initial definition of function addition to this simplified form highlights the power of algebraic manipulation. By carefully applying the rules of addition and combining like terms, we have transformed a symbolic expression into a concrete equation. This process is not just about finding the answer; it's about developing a deeper understanding of how functions work and how they can be combined. The result, 4x - 4, is not just a number; it's a function in its own right, with its own unique properties and characteristics. Understanding this is key to mastering the concept of function addition.

Conclusion

In this comprehensive exploration, we have successfully determined the simplest form of (f + g)(x) when f(x) = 3x + 1 and g(x) = x - 5. We began by defining function addition, (f + g)(x) = f(x) + g(x), and then applied this definition to the specific functions given. Through a step-by-step process of substitution and simplification, we arrived at the final answer:

(f + g)(x) = 4x - 4

This exercise demonstrates the fundamental principles of function addition and algebraic simplification. It underscores the importance of understanding definitions, applying them methodically, and paying attention to detail in each step. The ability to combine functions and simplify expressions is a crucial skill in mathematics, with applications in various fields, from physics and engineering to economics and computer science. By mastering these concepts, students and practitioners alike can tackle more complex problems and gain a deeper appreciation for the power and beauty of mathematics. Furthermore, this example serves as a template for solving similar problems involving other types of functions and operations. The key is to always start with the definition, substitute correctly, and simplify carefully. With practice and perseverance, anyone can master these skills and unlock the vast potential of mathematical thinking. The journey through this problem has not only provided an answer but also reinforced the importance of clear, logical thinking and the power of mathematical tools.