Function Composition An In-Depth Guide To Solving H(x) = F(x) * G(x)

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If f(x) = 11x + 9 and g(x) = 8x - 6, and h(x) is the product of f(x) and g(x), which equation defines h(x)?

In the realm of mathematics, functions serve as fundamental building blocks, and the operations we perform on them dictate the relationships they form. One such operation is the composition of functions, where we combine two or more functions to create a new function. This article delves into the concept of function composition, specifically focusing on the product of two functions, and provides a step-by-step guide to solving problems involving this operation.

Introduction to Function Composition: Unveiling the Product of Functions

At its core, function composition involves combining two functions, f(x) and g(x), to produce a new function, h(x). In the context of the product of functions, we define h(x) as the result of multiplying f(x) and g(x) together. Mathematically, this can be expressed as:

h(x) = f(x) * g(x)

This seemingly simple equation opens up a world of possibilities, allowing us to explore the intricate relationships between functions and their resulting products. To fully grasp the concept, let's delve into a specific example.

Illustrative Example: Constructing h(x) from f(x) and g(x)

Consider the following functions:

f(x) = 11x + 9 g(x) = 8x - 6

Our goal is to determine the equation that defines the function h(x), which is the product of f(x) and g(x). To achieve this, we'll follow a systematic approach, breaking down the process into manageable steps.

Step 1: Express h(x) as the Product of f(x) and g(x)

The initial step involves expressing h(x) as the product of f(x) and g(x), directly applying the definition of the product of functions:

h(x) = (11x + 9) * (8x - 6)

This step sets the stage for the subsequent expansion and simplification process.

Step 2: Expand the Product Using the Distributive Property

To simplify the expression, we employ the distributive property, which allows us to multiply each term in the first parenthesis by each term in the second parenthesis:

h(x) = 11x * (8x - 6) + 9 * (8x - 6)

Further expanding, we get:

h(x) = (11x * 8x) - (11x * 6) + (9 * 8x) - (9 * 6)

Step 3: Simplify the Expression by Combining Like Terms

Now, we simplify the expression by performing the multiplications and combining like terms:

h(x) = 88x² - 66x + 72x - 54

Combining the x terms, we arrive at:

h(x) = 88x² + 6x - 54

Therefore, the equation that defines the function h(x) is:

h(x) = 88x² + 6x - 54

Note: The original answer choices provided in the prompt had a slight error. The correct equation for h(x) is 88x² + 6x - 54, not 88x² + 3x - 54 as stated in option A. This highlights the importance of careful calculation and verification in mathematical problem-solving.

Key Concepts and Considerations in Function Composition

When dealing with function composition, it's crucial to keep in mind several key concepts and considerations:

  1. Order of Operations: The order in which you multiply the functions matters. In general, f(x) * g(x) is not the same as g(x) * f(x). However, in this specific case, multiplication is commutative, meaning the order does not affect the result.
  2. Domain and Range: The domain and range of the resulting function h(x) are influenced by the domains and ranges of the original functions f(x) and g(x). It's essential to consider these aspects when analyzing the behavior of the composite function.
  3. Generalization: The concept of function composition extends beyond the product of two functions. We can compose multiple functions together, or perform other operations like addition, subtraction, and division on functions.

Practical Applications of Function Composition

Function composition finds applications in various fields, including:

  • Calculus: Function composition is a fundamental concept in calculus, particularly in the chain rule for differentiation.
  • Computer Science: Function composition is used extensively in programming, where functions are often combined to create more complex operations.
  • Modeling: Function composition can be used to model real-world phenomena, where the output of one process serves as the input for another.

Solved Examples: Mastering Function Composition Techniques

To solidify your understanding of function composition, let's work through a few more examples:

Example 1:

Given:

f(x) = 2x - 1 g(x) = x² + 3

Find h(x) = f(x) * g(x).

Solution:

  1. h(x) = (2x - 1) * (x² + 3)
  2. h(x) = 2x * (x² + 3) - 1 * (x² + 3)
  3. h(x) = 2x³ + 6x - x² - 3
  4. h(x) = 2x³ - x² + 6x - 3

Example 2:

Given:

f(x) = √x g(x) = x + 2

Find h(x) = f(x) * g(x).

Solution:

  1. h(x) = √x * (x + 2)
  2. h(x) = x√x + 2√x

Practice Problems: Testing Your Understanding of Function Composition

To further enhance your skills, try solving these practice problems:

  1. Given f(x) = 3x + 2 and g(x) = x - 5, find h(x) = f(x) * g(x).
  2. Given f(x) = x² - 1 and g(x) = 2x + 1, find h(x) = f(x) * g(x).
  3. Given f(x) = 1/x and g(x) = x² + 1, find h(x) = f(x) * g(x).

Conclusion: Embracing the Power of Function Composition

Function composition, particularly the product of functions, is a powerful tool in mathematics, allowing us to create new functions from existing ones. By understanding the underlying principles and mastering the techniques involved, we can effectively solve problems and apply this concept in various fields. Through this comprehensive guide, you've gained a solid foundation in function composition, equipping you to tackle more complex mathematical challenges.

Remember: Practice is key to mastering any mathematical concept. Work through numerous examples and problems to solidify your understanding and build confidence in your abilities.

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