Solving Function Multiplication H(x) = F(x) * G(x)

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Introduction: Delving into Function Operations

In the realm of mathematics, particularly in algebra and calculus, functions are fundamental building blocks. Understanding how to manipulate and combine functions is crucial for solving complex problems. One common operation is the multiplication of functions, where we create a new function by multiplying the outputs of two existing functions. This article will explore this operation in detail, using a specific example to illustrate the process. We'll break down the steps, explain the underlying concepts, and highlight common pitfalls to avoid. This comprehensive guide aims to equip you with the knowledge and skills to confidently tackle function multiplication problems. We will go through the step-by-step solution of how to find h(x) when given f(x) and g(x) and h(x) is defined as the product of f(x) and g(x). This involves algebraic manipulation and a clear understanding of function notation. Function multiplication, while seemingly straightforward, lays the groundwork for more advanced concepts like composite functions and transformations. Therefore, mastering this skill is essential for any student venturing further into the world of mathematics. We'll also discuss how the result of function multiplication can differ significantly from simply adding or subtracting functions, emphasizing the importance of understanding the specific operation being performed. The principles discussed here can be applied to a wide range of functions, from simple linear equations to more complex polynomial expressions. By working through this example, you'll develop a solid foundation for dealing with function operations in general.

Problem Statement: Finding h(x) = f(x) * g(x)

Let's consider the following functions:

  • f(x) = 2x - 8
  • g(x) = -5x

Our goal is to find the function h(x), which is defined as the product of f(x) and g(x): h(x) = f(x) * g(x). This means we need to multiply the expressions representing f(x) and g(x) together. The problem presents us with two linear functions, f(x) and g(x), and asks us to determine the result of their product, h(x). This involves applying the distributive property of multiplication and simplifying the resulting expression. The ability to perform this operation is fundamental to understanding more complex function manipulations and is a key skill in algebra. This particular example serves as a good illustration because it involves both positive and negative terms, requiring careful attention to signs during the multiplication process. The solution will demonstrate how to systematically multiply the two expressions and combine like terms to arrive at the final expression for h(x). By understanding this process, you'll be well-prepared to tackle similar problems involving the multiplication of various types of functions. The problem also highlights the difference between function addition/subtraction and multiplication, emphasizing that each operation requires a different approach and can lead to significantly different results. Therefore, it's crucial to understand the specific operation being asked for in order to arrive at the correct solution. Finally, this problem underscores the importance of accurate algebraic manipulation and attention to detail when working with functions. A simple mistake in the multiplication or simplification process can lead to an incorrect answer.

Step-by-Step Solution: Multiplying the Functions

To find h(x), we substitute the expressions for f(x) and g(x) into the equation h(x) = f(x) * g(x):

  1. Write the equation: h(x) = (2x - 8) * (-5x)
  2. Apply the distributive property: We need to multiply each term in the first expression (2x - 8) by -5x. This means we multiply 2x by -5x and -8 by -5x. This step is crucial and is based on the fundamental distributive property of multiplication over addition (or subtraction). The distributive property states that a(b + c) = ab + ac. In our case, we're applying this property to the expression (-5x) * (2x - 8). It's essential to remember that each term inside the parentheses must be multiplied by the term outside. A common mistake is to only multiply the first term (2x) by -5x and forget to multiply the second term (-8). Therefore, it's helpful to write out each multiplication explicitly to avoid errors. This step sets the stage for simplifying the expression and arriving at the final answer for h(x). Proper application of the distributive property is a foundational skill in algebra, and this example provides a clear illustration of its use in the context of function multiplication. By carefully applying this property, we can expand the product of the two functions and prepare for the next step of combining like terms.
  3. Multiply the terms:
    • (2x) * (-5x) = -10x²
    • (-8) * (-5x) = 40x This step involves performing the actual multiplication of the individual terms. When multiplying terms with variables, remember to multiply the coefficients (the numbers in front of the variables) and add the exponents of the variables. In the first multiplication, (2x) * (-5x), we multiply 2 by -5 to get -10, and x by x to get x². Therefore, the result is -10x². In the second multiplication, (-8) * (-5x), we multiply -8 by -5 to get 40, and the x remains as it is. Thus, the result is 40x. It's crucial to pay attention to the signs during these multiplications. A negative times a negative is a positive, and a positive times a negative is a negative. A common mistake is to make an error with the signs, which can lead to an incorrect answer. This step highlights the importance of understanding the rules of exponents and the rules of multiplying signed numbers. Accuracy in these basic operations is essential for successfully manipulating algebraic expressions. By performing these multiplications correctly, we ensure that we have the correct terms to combine in the next step, leading to the final simplified expression for h(x).
  4. Combine the results: h(x) = -10x² + 40x

The Answer: Identifying the Correct Option

Comparing our result, h(x) = -10x² + 40x, with the given options, we find that option B is the correct answer.

  • A. h(x) = 10x² - 40x (Incorrect - signs are reversed)
  • B. h(x) = -10x² + 40x (Correct)
  • C. h(x) = -10x + 40 (Incorrect - this would be the result of addition/subtraction with incorrect sign handling)
  • D. h(x) = 10x - 40 (Incorrect - this would be the result of addition/subtraction) This step involves carefully examining the derived expression for h(x) and matching it with the provided options. It's crucial to pay close attention to both the coefficients and the signs of each term. A common mistake is to select an option that has the correct numbers but the wrong signs. This highlights the importance of double-checking your work and ensuring that all signs are correct. Option A is incorrect because it has the correct coefficients but the signs are reversed. Options C and D are incorrect because they do not have the x² term, indicating a misunderstanding of the multiplication process. They might represent the result of adding or subtracting the functions f(x) and g(x) in some way, but they do not represent the product. Therefore, by carefully comparing the derived expression with the options, we can confidently identify option B as the correct answer. This step reinforces the importance of accuracy in algebraic manipulation and the ability to interpret the results in the context of the given problem.

Key Takeaways: Mastering Function Multiplication

  • The distributive property is key: Remember to multiply each term in one function by each term in the other function.
  • Pay attention to signs: Be careful with negative signs, as they can easily lead to errors.
  • Combine like terms: Simplify the expression by combining terms with the same variable and exponent.
  • Function multiplication creates new functions: The resulting function, h(x), has different properties and behavior compared to the original functions, f(x) and g(x). These takeaways summarize the essential concepts and steps involved in function multiplication. The distributive property is the cornerstone of this operation, allowing us to expand the product of two functions into a sum of terms. However, the distributive property must be applied correctly by multiplying every term in one function by every term in the other. Neglecting to multiply a term or making a mistake with signs is a common source of errors. This highlights the importance of careful and systematic application of the property. Paying close attention to signs is equally crucial. Negative signs can easily be overlooked or mishandled, leading to incorrect results. Developing a habit of double-checking the signs of each term during the multiplication and simplification process is essential. Combining like terms is the final step in simplifying the expression for the product function. This involves identifying terms with the same variable and exponent and adding their coefficients. Failing to combine like terms will leave the expression in an unsimplified form. The most important takeaway is that function multiplication creates a new function with different properties and behavior compared to the original functions. The product function h(x) may have a different domain, range, and graph compared to f(x) and g(x). Understanding this difference is crucial for applying function multiplication in more advanced mathematical contexts. By keeping these key takeaways in mind, you can confidently approach and solve function multiplication problems.

Practice Problems: Enhancing Your Understanding

To further solidify your understanding of function multiplication, try these practice problems:

  1. If f(x) = x + 3 and g(x) = 2x - 1, find h(x) = f(x) * g(x).
  2. If f(x) = -3x + 5 and g(x) = x - 2, find h(x) = f(x) * g(x).
  3. If f(x) = x² and g(x) = x + 4, find h(x) = f(x) * g(x). These practice problems provide an opportunity to apply the concepts and steps discussed in this article. Working through these problems will help you identify any areas where you may need further clarification or practice. Problem 1 involves the multiplication of two linear functions, similar to the example problem. This will reinforce your understanding of the distributive property and the process of combining like terms. Problem 2 also involves linear functions, but with negative coefficients. This will further test your ability to handle signs correctly during the multiplication process. Problem 3 introduces a quadratic function (x²) as one of the factors. This will require you to apply the distributive property and the rules of exponents in a slightly more complex setting. By attempting these problems, you'll not only strengthen your understanding of function multiplication but also develop your problem-solving skills in algebra. It's recommended to write out each step of the solution clearly and carefully, paying attention to signs and coefficients. After completing the problems, you can check your answers and identify any mistakes. If you encounter difficulties, review the explanation and examples in this article or consult additional resources. The key to mastering function multiplication is consistent practice and a clear understanding of the underlying principles. These problems provide a valuable opportunity to build your confidence and proficiency in this essential mathematical skill.

Conclusion: The Power of Function Operations

Function multiplication is a fundamental operation in mathematics that allows us to create new functions with unique properties. By understanding the distributive property and carefully applying the steps outlined in this article, you can confidently solve problems involving the multiplication of functions. Mastering function operations is essential for success in algebra, calculus, and other advanced mathematical fields. This article has provided a comprehensive guide to function multiplication, including a step-by-step solution to a specific example, key takeaways, and practice problems. By working through this material, you have gained a solid foundation in this essential mathematical concept. Function multiplication is not just a mechanical process; it's a powerful tool for creating and manipulating mathematical models. Understanding how to combine functions in different ways allows us to represent complex relationships and phenomena in a concise and elegant manner. The principles discussed in this article extend beyond simple polynomial functions and can be applied to a wide range of function types, including trigonometric, exponential, and logarithmic functions. As you continue your mathematical journey, you will encounter function multiplication in various contexts. Whether you're solving equations, graphing functions, or analyzing data, the ability to multiply functions effectively will be a valuable asset. Therefore, it's worthwhile to invest the time and effort to master this skill. Remember to practice regularly, review the key takeaways, and seek help when needed. With dedication and perseverance, you can confidently tackle any function multiplication problem and unlock the power of function operations in mathematics.