Function Subtraction Explained Finding (f-g)(x)

In the realm of mathematics, particularly within the study of functions, one fundamental operation is the subtraction of functions. Given two functions, f(x) and g(x), their difference, denoted as (f-g)(x), is a new function formed by subtracting the values of g(x) from f(x) for every value of x. This operation is crucial for various applications, such as modeling scenarios where one quantity is reduced or offset by another. In this comprehensive exploration, we will delve into the process of subtracting functions, providing a step-by-step guide and clarifying the underlying principles. We will tackle the specific problem of finding (f-g)(x) when f(x) = -x^2 + 6x - 1 and g(x) = 3x^2 - 4x - 1, illustrating the method with a concrete example. Through this detailed explanation, you will gain a solid understanding of function subtraction, equipping you with the skills to handle similar problems confidently and accurately.

The Concept of Function Subtraction

Function subtraction is a core concept in algebra and calculus, extending the familiar arithmetic operation of subtraction to functions. To truly grasp function subtraction, think of functions as machines: f(x) takes an input x and produces an output based on its rule, and g(x) does the same, but possibly with a different rule. When we subtract g(x) from f(x), we are essentially creating a new machine that takes the same input x, calculates both f(x) and g(x) separately, and then subtracts the result of g(x) from f(x). The result of this subtraction is the output of the new function, (f-g)(x). This concept is vital because it allows us to compare and contrast the behavior of two functions, to model situations where one function's effect diminishes another's, or to simply create new functions with altered characteristics. The notation (f-g)(x) is a concise way of representing this operation, making it easy to express and manipulate function differences in mathematical expressions and problem-solving scenarios. Understanding this concept thoroughly lays the groundwork for more advanced topics in function analysis and calculus.

Steps to Calculate (f-g)(x)

Calculating (f-g)(x) is a straightforward process that involves a few key steps. The first crucial step is to write down the expressions for both f(x) and g(x) explicitly. This step ensures that you have a clear understanding of the functions you are working with and reduces the likelihood of errors in subsequent steps. Next, express the subtraction (f-g)(x) as f(x) - g(x). This notational shift emphasizes that you are subtracting the entire function g(x) from f(x). The third step, and often the most critical, is to substitute the expressions for f(x) and g(x) into the equation. Remember to enclose g(x) in parentheses, as this ensures that the negative sign is properly distributed across all terms within g(x). After substitution, the next step involves distributing the negative sign across the terms in g(x). This is a crucial step, as failing to distribute the negative sign correctly can lead to significant errors in the final result. Once the negative sign has been distributed, combine like terms, meaning terms with the same power of x. This simplification step involves adding or subtracting the coefficients of the like terms. Finally, write the resulting expression in its simplest form. This usually involves arranging the terms in descending order of the powers of x. By following these steps meticulously, you can accurately calculate (f-g)(x) for any given functions f(x) and g(x).

Applying the Steps to the Given Problem

To effectively illustrate the calculation of (f-g)(x), let's apply the steps outlined above to the specific problem where f(x) = -x^2 + 6x - 1 and g(x) = 3x^2 - 4x - 1. First, we write down the expressions for the functions: f(x) = -x^2 + 6x - 1 and g(x) = 3x^2 - 4x - 1. The second step is to express (f-g)(x) as f(x) - g(x), which clarifies the operation we are about to perform. Next, we substitute the expressions for f(x) and g(x) into the equation, being careful to enclose g(x) in parentheses: (f-g)(x) = (-x^2 + 6x - 1) - (3x^2 - 4x - 1). This is a critical step as the parentheses ensure the correct distribution of the negative sign. The next step involves distributing the negative sign across the terms in g(x): (f-g)(x) = -x^2 + 6x - 1 - 3x^2 + 4x + 1. Now, we combine like terms. We have -x^2 and -3x^2, which combine to -4x^2. Next, we combine 6x and 4x, resulting in 10x. Finally, we combine the constants -1 and +1, which cancel each other out, leaving 0. Thus, we have (f-g)(x) = -4x^2 + 10x + 0. The final step is to write the resulting expression in its simplest form, which is (f-g)(x) = -4x^2 + 10x. This detailed application of the steps demonstrates how to systematically calculate (f-g)(x), ensuring accuracy and clarity in the solution.

Detailed Solution for f(x) - g(x)

Now, let's work through the problem step-by-step to solidify your understanding of function subtraction. We are given f(x) = -x^2 + 6x - 1 and g(x) = 3x^2 - 4x - 1. Our goal is to find (f-g)(x), which means we need to subtract the function g(x) from the function f(x). We begin by writing down the expressions for the functions: f(x) = -x^2 + 6x - 1 and g(x) = 3x^2 - 4x - 1. The next step is to set up the subtraction: (f-g)(x) = f(x) - g(x). Then, we substitute the given expressions for f(x) and g(x) into the equation. It's crucial to use parentheses around g(x) to ensure we subtract the entire function: (f-g)(x) = (-x^2 + 6x - 1) - (3x^2 - 4x - 1). The next key step is to distribute the negative sign across the terms inside the parentheses of g(x). This means changing the sign of each term in g(x): (f-g)(x) = -x^2 + 6x - 1 - 3x^2 + 4x + 1. Now, we combine like terms. We identify the x^2 terms: -x^2 and -3x^2. Adding these together gives us -4x^2. Next, we identify the x terms: 6x and 4x. Adding these together gives us 10x. Finally, we identify the constant terms: -1 and +1. These cancel each other out, resulting in 0. So, we have (f-g)(x) = -4x^2 + 10x + 0. Simplifying, we get (f-g)(x) = -4x^2 + 10x. Thus, the result of subtracting g(x) from f(x) is the quadratic function -4x^2 + 10x. This step-by-step breakdown ensures that you can follow the logic and accurately perform function subtraction in similar problems.

Correct Answer and Explanation

After meticulously performing the function subtraction, we have arrived at the solution: (f-g)(x) = -4x^2 + 10x. This result corresponds to option D in the given choices. Therefore, the correct answer is D. (f-g)(x) = -4x^2 + 10x. To reiterate, we found this solution by first substituting the expressions for f(x) and g(x) into the equation (f-g)(x) = f(x) - g(x). We then distributed the negative sign across the terms of g(x), ensuring that each term was correctly subtracted. This step is crucial to avoid errors in the calculation. After distributing the negative sign, we combined like terms, which involved adding the coefficients of terms with the same power of x. This simplification process led us to the final expression, -4x^2 + 10x. This detailed explanation reinforces the accuracy of our solution and highlights the importance of each step in the process. By understanding the rationale behind each step, you can confidently apply this method to other function subtraction problems. This comprehensive approach ensures not just the correct answer but also a deeper understanding of the underlying mathematical principles.

Common Mistakes to Avoid

When performing function subtraction, there are several common mistakes that students often make. Recognizing these pitfalls and actively avoiding them is crucial for achieving accuracy and mastering the concept. One of the most frequent errors is failing to distribute the negative sign correctly when subtracting g(x) from f(x). Remember that the entire function g(x) is being subtracted, so the negative sign must be applied to every term within g(x). Forgetting to do this can lead to incorrect signs and a wrong answer. Another common mistake is combining unlike terms. Only terms with the same variable and exponent can be combined. For example, x^2 terms can only be combined with other x^2 terms, and x terms can only be combined with other x terms. Mixing these up will result in an incorrect expression. A third common error is making mistakes in basic arithmetic, such as adding or subtracting coefficients incorrectly. It's essential to double-check your calculations to avoid these simple mistakes. A helpful strategy is to write out each step clearly and carefully, paying attention to the signs and coefficients. Another useful tip is to use parentheses to maintain clarity and organization, especially when dealing with multiple terms and negative signs. By being aware of these common mistakes and taking proactive steps to avoid them, you can significantly improve your accuracy and confidence in solving function subtraction problems. Practicing with a variety of examples and reviewing your work can further help solidify your understanding and prevent these errors.

Practice Problems

To further solidify your understanding of function subtraction, it's beneficial to work through additional practice problems. These exercises will allow you to apply the steps and concepts we've discussed, reinforcing your skills and building confidence. Here are a few practice problems to get you started:

1. If f(x) = 2x^2 - 3x + 1 and g(x) = x^2 + 5x - 3, find (f-g)(x).
2. Given f(x) = -3x^3 + 4x - 2 and g(x) = 2x^3 - x^2 + 1, determine (f-g)(x).
3. Let f(x) = 5x + 7 and g(x) = -2x + 3. Calculate (f-g)(x).
4. If f(x) = x^4 - 2x^2 + 5 and g(x) = x^4 + x^2 - 2, find (f-g)(x).

For each problem, follow the steps we outlined earlier: first, write down the expressions for f(x) and g(x); second, set up the subtraction (f-g)(x) = f(x) - g(x); third, substitute the expressions for the functions, using parentheses around g(x); fourth, distribute the negative sign across the terms in g(x); fifth, combine like terms; and finally, write the resulting expression in its simplest form. After completing these problems, review your work and compare your solutions with the correct answers. If you encounter any difficulties, revisit the steps and explanations provided earlier in this discussion. Working through these practice problems will not only enhance your understanding of function subtraction but also improve your problem-solving skills in mathematics more broadly. Consistent practice is key to mastering any mathematical concept, and function subtraction is no exception.

Conclusion

In conclusion, the subtraction of functions, represented as (f-g)(x), is a fundamental operation in mathematics that allows us to create new functions by finding the difference between two given functions. This process involves several key steps: writing down the expressions for f(x) and g(x), expressing (f-g)(x) as f(x) - g(x), substituting the function expressions, distributing the negative sign across the terms of g(x), combining like terms, and simplifying the resulting expression. By following these steps meticulously, you can accurately calculate (f-g)(x) for any given functions. We demonstrated this process with the specific example of f(x) = -x^2 + 6x - 1 and g(x) = 3x^2 - 4x - 1, ultimately finding that (f-g)(x) = -4x^2 + 10x. We also discussed common mistakes to avoid, such as failing to distribute the negative sign correctly and combining unlike terms. To further enhance your understanding, we provided practice problems that allow you to apply the concepts and techniques discussed. Mastering function subtraction is crucial for success in algebra and calculus, as it forms the basis for more advanced topics and applications. By consistently practicing and reviewing these concepts, you can build a solid foundation in function operations and improve your overall mathematical proficiency. The ability to confidently perform function subtraction will undoubtedly prove valuable in your mathematical studies and beyond.