Solving (f-g)(x) Understanding Function Subtraction

In the realm of mathematics, particularly in algebra, functions play a crucial role. They serve as a fundamental tool for modeling relationships between variables and understanding how changes in one variable affect another. Among the various operations that can be performed on functions, subtraction is a common and essential one. In this article, we will delve into the process of subtracting functions, specifically focusing on solving for (f-g)(x). This operation involves taking two functions, f(x) and g(x), and finding the difference between them. Understanding how to perform this operation is vital for various mathematical applications, including calculus, data analysis, and more. Let's embark on this journey of exploring function subtraction and unravel the steps involved in solving for (f-g)(x). By the end of this discussion, you'll have a solid grasp of the concept and be able to confidently tackle problems involving function subtraction.

Defining Functions f(x) and g(x)

Before we dive into the process of subtracting functions, it's essential to clearly define the functions we'll be working with. Let's consider two functions, f(x) and g(x), which serve as the foundation for our exploration of function subtraction. Imagine f(x) as a mathematical machine that takes an input x and transforms it according to a specific rule. For instance, f(x) could be a simple function like f(x) = x^2 + 2x, where the input x is squared and then added to twice its value. Similarly, g(x) represents another mathematical machine that operates on the input x based on its own unique rule. It could be a function like g(x) = x - 4, where the input x is simply reduced by 4. These functions, f(x) and g(x), are the building blocks upon which we'll perform the subtraction operation to find (f-g)(x). Understanding the nature of these functions and their respective rules is crucial for comprehending the subsequent steps in the subtraction process. So, let's keep these functions in mind as we proceed to the next stage of our exploration.

Understanding the Subtraction of Functions: (f-g)(x)

Now that we've established our functions, f(x) and g(x), let's delve into the core concept of subtracting functions, denoted as (f-g)(x). This operation represents the difference between the values of f(x) and g(x) for a given input x. In simpler terms, it involves taking the output of f(x) and subtracting the output of g(x) from it. Mathematically, we can express this as (f-g)(x) = f(x) - g(x). This equation forms the foundation for our understanding of function subtraction. To illustrate this concept further, let's consider our previously defined functions, f(x) = x^2 + 2x and g(x) = x - 4. When we subtract g(x) from f(x), we're essentially finding the difference between their respective expressions. This subtraction process is not merely a symbolic operation; it represents a fundamental mathematical relationship between the two functions. It allows us to analyze how the functions interact with each other and how their outputs differ for various values of x. Understanding this subtraction operation is crucial for various mathematical applications, from solving equations to modeling real-world phenomena. So, let's keep this principle in mind as we move forward to the next step, where we'll apply it to solve for (f-g)(x) in a specific scenario.

Step-by-Step Solution: Calculating (f-g)(x)

Now, let's put our understanding of function subtraction into practice and walk through the step-by-step solution for calculating (f-g)(x). We'll continue to use our previously defined functions, f(x) = x^2 + 2x and g(x) = x - 4, to illustrate the process. Remember, the formula for (f-g)(x) is f(x) - g(x). So, our first step is to substitute the expressions for f(x) and g(x) into this formula. This gives us (f-g)(x) = (x^2 + 2x) - (x - 4). Notice how we've enclosed the expressions in parentheses to ensure that we subtract the entire expression for g(x). Next, we need to simplify the expression by distributing the negative sign to the terms inside the second set of parentheses. This means we multiply each term in (x - 4) by -1. Doing so, we get (f-g)(x) = x^2 + 2x - x + 4. Now, we can combine like terms to further simplify the expression. In this case, we have two terms with x: 2x and -x. Combining these terms, we get x. So, our expression becomes (f-g)(x) = x^2 + x + 4. And there you have it! We've successfully calculated (f-g)(x) for the given functions f(x) and g(x). This step-by-step process demonstrates the methodical approach to subtracting functions, ensuring that we arrive at the correct result. As we move forward, remember that this process can be applied to any pair of functions, making it a versatile tool in your mathematical arsenal.

Analyzing the Result: Understanding the New Function

With the calculation of (f-g)(x) completed, we've arrived at a new function, x^2 + x + 4. But what does this new function tell us? Analyzing the result is a crucial step in understanding the implications of subtracting functions. This new function, (f-g)(x), represents the difference between the original functions, f(x) and g(x), for every value of x. In essence, it captures the vertical distance between the graphs of f(x) and g(x) at each point. Let's delve deeper into the characteristics of this new function. Notice that (f-g)(x) = x^2 + x + 4 is a quadratic function, meaning it has the form ax^2 + bx + c. Quadratic functions are known for their parabolic shape when graphed, and this one is no exception. The coefficient of the x^2 term, which is 1 in this case, determines the direction in which the parabola opens. Since it's positive, the parabola opens upwards, indicating that the function has a minimum value. The other coefficients, b and c, influence the position and shape of the parabola. In this specific function, the x term and the constant term shift the parabola vertically and horizontally, respectively. Understanding the nature of this new function allows us to make predictions about its behavior and how it relates to the original functions. For instance, we can determine the minimum value of (f-g)(x) and the values of x at which it occurs. This analysis is invaluable in various applications, from optimization problems to modeling real-world scenarios. So, as we conclude this section, remember that the result of subtracting functions is not just a mathematical expression; it's a new function with its own unique characteristics and properties.

Practical Applications of Function Subtraction

Now that we've mastered the mechanics of subtracting functions and analyzing the resulting function, let's explore some practical applications of this mathematical operation. Function subtraction isn't just an abstract concept; it has real-world implications in various fields, from economics to physics. One common application is in cost analysis. Imagine a company that has two different production processes for manufacturing a product. Each process has its own cost function, representing the cost of producing a certain quantity of goods. By subtracting the cost functions, the company can determine the difference in cost between the two processes for any given production level. This information can then be used to make informed decisions about which process is more cost-effective. Another application is in physics, particularly in the study of motion. Suppose we have two objects moving along a path, each with its own position function describing its location at a given time. By subtracting the position functions, we can determine the relative distance between the objects at any point in time. This can be useful for analyzing the motion of objects, such as cars on a highway or planets in orbit. In signal processing, function subtraction is used to remove unwanted noise from a signal. If we have a signal that is contaminated with noise, we can subtract a noise signal from the original signal to obtain a cleaner version. This technique is widely used in audio and image processing to improve the quality of recordings and images. These are just a few examples of the many practical applications of function subtraction. This operation provides a powerful tool for comparing and analyzing functions in various contexts, making it an essential concept in mathematics and its applications.

Real-World Example: Profit Calculation

To further illustrate the practical applications of function subtraction, let's consider a real-world example involving profit calculation. Imagine you're running a small business that sells handmade crafts. Your revenue, the total income you generate from selling your crafts, can be represented by a function, R(x), where x is the number of crafts you sell. Similarly, your cost, the expenses you incur in producing your crafts, can be represented by another function, C(x), where x is again the number of crafts you produce. Your profit, the amount of money you actually make after deducting your costs from your revenue, is the difference between your revenue and your cost. This is where function subtraction comes into play. The profit function, P(x), can be expressed as P(x) = R(x) - C(x). By subtracting the cost function from the revenue function, you can determine your profit for any given number of crafts sold. Let's say your revenue function is R(x) = 25x, meaning you earn $25 for each craft you sell. And your cost function is C(x) = 10x + 50, where $10x represents the cost of materials and labor for each craft, and $50 represents your fixed costs, such as rent and utilities. To find your profit function, we subtract C(x) from R(x): P(x) = R(x) - C(x) = 25x - (10x + 50). Simplifying this expression, we get P(x) = 15x - 50. This profit function tells you how much profit you'll make for selling a certain number of crafts. For example, if you sell 10 crafts, your profit would be P(10) = 15(10) - 50 = $100. This real-world example demonstrates how function subtraction can be used to solve practical problems in business and finance. By understanding the relationship between revenue, cost, and profit, you can make informed decisions about pricing, production levels, and overall business strategy.

Conclusion

In this comprehensive exploration, we've delved into the world of function subtraction, unraveling its intricacies and showcasing its practical significance. We began by defining functions f(x) and g(x), setting the stage for our understanding of function operations. We then focused on the core concept of subtracting functions, represented as (f-g)(x), emphasizing that it involves finding the difference between the values of f(x) and g(x) for a given input x. Through a step-by-step solution, we demonstrated how to calculate (f-g)(x), ensuring a clear understanding of the process. Analyzing the result, we discovered that (f-g)(x) represents a new function with its own unique characteristics and properties, such as its parabolic shape and minimum value. We further highlighted the practical applications of function subtraction in various fields, including cost analysis, physics, and signal processing. To solidify our understanding, we examined a real-world example of profit calculation, illustrating how function subtraction can be used to model and solve problems in business and finance. By subtracting the cost function from the revenue function, we were able to determine the profit function, providing valuable insights into the business's financial performance. As we conclude this discussion, it's evident that function subtraction is not just a mathematical operation; it's a powerful tool for analyzing relationships between functions and solving real-world problems. Whether you're calculating profits, analyzing motion, or processing signals, the principles of function subtraction can provide valuable insights and solutions. So, embrace this mathematical concept and apply it to the world around you, and you'll be amazed at its versatility and power.

Therefore, based on the step-by-step solution provided above, the correct answer is B. x^2 + x + 4