Solving For H(x) Understanding Function Subtraction With F(x) And G(x)

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In the realm of mathematics, particularly in algebra and calculus, understanding functions and their operations is crucial. One fundamental operation is the subtraction of functions. This article delves into the concept of function subtraction, providing a comprehensive explanation and a step-by-step solution to a specific problem. We'll explore the process of finding a new function, h(x), which is defined as the difference between two given functions, f(x) and g(x). This exploration will not only enhance your understanding of function operations but also equip you with the skills to tackle similar problems.

Function Subtraction: The Basics

Function subtraction is a straightforward operation where we subtract one function from another. Given two functions, f(x) and g(x), their difference, denoted as h(x), is defined as:

h(x) = f(x) - g(x)

This means that for any given value of x, we find the corresponding values of f(x) and g(x), and then subtract the value of g(x) from the value of f(x). The result is the value of the new function h(x) at that particular x. Understanding this fundamental concept is key to solving problems involving function subtraction.

To further clarify, let's consider a scenario. Imagine f(x) represents the cost of producing x items, and g(x) represents the revenue generated from selling x items. Then, h(x) = f(x) - g(x) would represent the profit (or loss) from producing and selling x items. This real-world analogy helps to contextualize the mathematical operation and make it more relatable.

When performing function subtraction, it's crucial to pay close attention to the order of operations and the signs of the terms involved. A simple mistake in sign manipulation can lead to an incorrect result. Therefore, a systematic and meticulous approach is essential. We will demonstrate this approach in detail as we solve the problem presented in this article.

Problem Statement: Finding h(x)

The core of this article is to solve a specific problem involving function subtraction. We are given two functions:

  • f(x) = 8x - 2
  • g(x) = 9x + 2

Our objective is to find the function h(x), which is defined as the difference between f(x) and g(x):

h(x) = f(x) - g(x)

This problem is a classic example of function subtraction and provides a clear illustration of the steps involved in the process. The solution will involve substituting the given expressions for f(x) and g(x) into the equation for h(x) and then simplifying the resulting expression. This process will highlight the importance of algebraic manipulation and attention to detail.

Before we dive into the solution, it's worth noting the significance of this type of problem. Function subtraction, like other function operations, is a building block for more advanced mathematical concepts. It's used extensively in calculus, where we analyze the rates of change of functions, and in various applications of mathematics in science and engineering. Therefore, mastering this fundamental operation is crucial for a solid foundation in mathematics.

Step-by-Step Solution: Calculating h(x)

Now, let's embark on the step-by-step solution to find h(x). We'll meticulously walk through each step, ensuring clarity and understanding.

Step 1: Substitute the expressions for f(x) and g(x) into the equation for h(x).

We know that h(x) = f(x) - g(x). We are also given that f(x) = 8x - 2 and g(x) = 9x + 2. Substituting these expressions into the equation for h(x), we get:

h(x) = (8x - 2) - (9x + 2)

This step is crucial as it sets the stage for the subsequent algebraic manipulation. It's important to enclose the expressions for f(x) and g(x) in parentheses to maintain the correct order of operations, especially when dealing with subtraction.

Step 2: Distribute the negative sign.

The next step involves distributing the negative sign in front of the parentheses containing the expression for g(x). This is a critical step, as failing to distribute the negative sign correctly is a common source of error. Remember that subtracting a quantity is the same as adding its negative.

Distributing the negative sign, we get:

h(x) = 8x - 2 - 9x - 2

Notice how both terms inside the parentheses, 9x and +2, have their signs changed after the negative sign is distributed. This is a direct application of the distributive property of multiplication over addition and subtraction.

Step 3: Combine like terms.

The final step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, the like terms are 8x and -9x, and the constant terms are -2 and -2. Combining like terms simplifies the expression and brings us closer to the final answer.

Combining the x terms, we have 8x - 9x = -x. Combining the constant terms, we have -2 - 2 = -4. Therefore, the simplified expression for h(x) is:

h(x) = -x - 4

This is our final answer. We have successfully found the function h(x) by subtracting g(x) from f(x).

Solution and Answer Choices

Therefore, based on our step-by-step solution, the correct answer is:

C. h(x) = -x - 4

This matches one of the answer choices provided in the problem statement. Let's briefly analyze why the other answer choices are incorrect. This will further solidify our understanding of the solution process.

  • A. h(x) = 17x + 14: This answer likely results from incorrectly adding f(x) and g(x) instead of subtracting them, and also making errors with the signs.
  • B. h(x) = x + 4: This answer could be a result of subtracting f(x) from g(x) instead of g(x) from f(x), or making errors with the signs during the subtraction process.
  • D. h(x) = -x + 14: This answer likely has the correct x term but an incorrect constant term, possibly due to a sign error when combining the constant terms.

By understanding the errors that lead to these incorrect answers, we reinforce our understanding of the correct method and the importance of paying attention to detail.

Conclusion: Mastering Function Subtraction

In conclusion, this article has provided a comprehensive guide to understanding and performing function subtraction. We started with the basic definition of function subtraction, explored a real-world analogy to contextualize the operation, and then delved into a step-by-step solution of a specific problem. By meticulously working through each step, we demonstrated the importance of algebraic manipulation and attention to detail.

The key takeaways from this article are:

  • Function subtraction involves subtracting one function from another: h(x) = f(x) - g(x).
  • Substituting the expressions for f(x) and g(x) into the equation for h(x) is the first step.
  • Distributing the negative sign correctly is crucial to avoid errors.
  • Combining like terms simplifies the expression and leads to the final answer.
  • Understanding the potential errors that lead to incorrect answers reinforces the correct method.

Mastering function subtraction is a fundamental skill in mathematics. It's a building block for more advanced concepts in algebra, calculus, and various applications of mathematics in science and engineering. By practicing and applying the principles outlined in this article, you can confidently tackle problems involving function subtraction and build a solid foundation in mathematics. Remember, a systematic and meticulous approach, combined with a clear understanding of the underlying concepts, is the key to success in mathematics.

This article has hopefully demystified the process of function subtraction and provided you with the tools and knowledge to solve similar problems. Keep practicing, and you'll become proficient in this essential mathematical operation.