Multiplying Functions A Step By Step Guide To Finding H(x)

In the realm of mathematics, particularly within the study of functions, operations extend beyond the familiar arithmetic manipulations of addition, subtraction, multiplication, and division applied to numerical values. Functions themselves can be subjected to these operations, leading to the creation of new functions with unique properties and behaviors. This article aims to dissect one such operation: the multiplication of functions. We will meticulously explore the process of multiplying two given functions, f(x) and g(x), to obtain a new function, h(x). This exploration will not only provide a step-by-step guide to the multiplication process but also emphasize the underlying principles that govern function operations. By understanding these principles, we can gain a deeper appreciation for the elegance and versatility of mathematical functions.

The specific problem we will address involves two linear functions: f(x) = 6x + 7 and g(x) = 4x - 2. Our objective is to determine the product of these functions, denoted as h(x) = f(x) * g(x). This seemingly straightforward task requires careful application of the distributive property and a keen eye for combining like terms. The result, h(x), will reveal the nature of the function resulting from the multiplication of two linear functions. This exploration is crucial for anyone seeking to master the fundamentals of function operations and their applications in various mathematical contexts. Let's embark on this journey of mathematical discovery, unraveling the intricacies of function multiplication and solidifying our understanding of this essential concept.

The function multiplication process extends beyond mere symbolic manipulation; it provides insights into how functions interact and influence each other's behavior. When we multiply f(x) and g(x), we are essentially creating a new function whose output at any given x is the product of the outputs of f(x) and g(x) at that same x. This concept has profound implications in various fields, such as physics, engineering, and economics, where functions are used to model real-world phenomena. Understanding function multiplication allows us to create more complex models that capture the interactions between different variables and their effects on the overall system.

Before we embark on the solution, let's formally restate the problem at hand. We are given two functions:

• f(x) = 6x + 7
• g(x) = 4x - 2

Our goal is to determine the function h(x), which is defined as the product of f(x) and g(x). In mathematical notation, this is expressed as:

• h(x) = f(x) * g(x)

This seemingly simple equation encapsulates the core of the problem. To find h(x), we must substitute the expressions for f(x) and g(x) and then perform the multiplication. This process involves applying the distributive property, a fundamental algebraic principle that governs how expressions within parentheses are multiplied. Mastering this step is crucial for accurately determining the expression for h(x). Moreover, we need to simplify the resulting expression by combining like terms. This involves identifying terms with the same variable and exponent and then adding their coefficients. This simplification process ensures that the final expression for h(x) is in its most concise and understandable form.

Now, let's delve into the step-by-step solution to determine h(x). We begin by substituting the given expressions for f(x) and g(x) into the equation:

• h(x) = (6x + 7) * (4x - 2)

Next, we apply the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last), to multiply the two binomials. This involves multiplying each term in the first binomial by each term in the second binomial:

1. First: Multiply the first terms of each binomial: (6x) * (4x) = 24x²
2. Outer: Multiply the outer terms of the binomials: (6x) * (-2) = -12x
3. Inner: Multiply the inner terms of the binomials: (7) * (4x) = 28x
4. Last: Multiply the last terms of each binomial: (7) * (-2) = -14

Combining these results, we obtain:

• h(x) = 24x² - 12x + 28x - 14

Finally, we simplify the expression by combining like terms. The like terms in this expression are -12x and 28x. Adding their coefficients, we get:

• -12x + 28x = 16x

Substituting this back into the expression for h(x), we arrive at the final answer:

• h(x) = 24x² + 16x - 14

This meticulous step-by-step process highlights the importance of the distributive property and the careful combination of like terms in function multiplication. By breaking down the problem into smaller, manageable steps, we can ensure accuracy and avoid common errors. This approach not only leads to the correct answer but also enhances our understanding of the underlying mathematical principles.

Having determined that h(x) = 24x² + 16x - 14, it's crucial to analyze the result and understand the nature of the function we have obtained. The expression for h(x) is a quadratic function, characterized by the presence of a term with x². This observation is significant because it tells us that the graph of h(x) will be a parabola, a U-shaped curve that opens either upwards or downwards depending on the sign of the coefficient of the x² term. In this case, the coefficient is 24, which is positive, indicating that the parabola opens upwards.

The quadratic nature of h(x) arises from the multiplication of two linear functions. Each linear function, f(x) and g(x), represents a straight line. When we multiply them, we introduce a quadratic term, which fundamentally alters the shape of the resulting function. This transformation from linear to quadratic behavior is a key concept in understanding function operations.

Furthermore, we can analyze the coefficients of the quadratic function to gain insights into its properties. The coefficient of the x² term (24) determines the parabola's curvature – a larger coefficient implies a steeper curve. The coefficient of the x term (16) influences the parabola's position along the x-axis, and the constant term (-14) represents the y-intercept, the point where the parabola intersects the y-axis. By understanding these relationships, we can sketch a rough graph of h(x) and predict its behavior without explicitly plotting numerous points.

The quadratic function h(x) also possesses a vertex, which is the point where the parabola changes direction. The vertex represents either the minimum or maximum value of the function, depending on whether the parabola opens upwards or downwards. In this case, since the parabola opens upwards, the vertex represents the minimum value of h(x). Finding the vertex involves using specific formulas or techniques, which further enhance our understanding of the function's characteristics. This analysis of the result not only confirms the correctness of our calculations but also deepens our comprehension of quadratic functions and their graphical representations.

Now that we have meticulously derived the expression for h(x), let's evaluate the given answer choices to identify the correct option. We found that h(x) = 24x² + 16x - 14. Comparing this result with the provided options:

A. h(x) = 24x² - 14 B. h(x) = 24x² + 40x + 14 C. h(x) = 24x² + 16x - 14 D. h(x) = 24x² + 12x - 14

Clearly, option C matches our calculated result perfectly. The other options differ in the coefficient of the x term or the constant term, highlighting the importance of careful calculation and simplification. This step of evaluating the answer choices serves as a crucial verification process, ensuring that we have not made any errors in our calculations.

Furthermore, this comparison underscores the importance of understanding the underlying mathematical concepts. By correctly applying the distributive property and simplifying the expression, we were able to confidently identify the correct answer. This approach, rooted in a solid understanding of mathematical principles, is far more effective than relying on guesswork or pattern recognition. The ability to evaluate answer choices critically and select the correct option based on sound mathematical reasoning is a valuable skill in problem-solving and beyond.

In conclusion, we have successfully determined the function h(x), defined as the product of f(x) = 6x + 7 and g(x) = 4x - 2. Through a meticulous step-by-step process, we applied the distributive property, combined like terms, and arrived at the expression h(x) = 24x² + 16x - 14. This exercise has not only provided us with the solution to the specific problem but also reinforced our understanding of function multiplication and the properties of quadratic functions.

Function multiplication is a fundamental operation in mathematics, with applications spanning various fields. By mastering this concept, we equip ourselves with a powerful tool for modeling and analyzing real-world phenomena. The process involves careful application of algebraic principles and a keen eye for detail. From the distributive property to the combination of like terms, each step plays a crucial role in arriving at the correct answer.

Furthermore, the analysis of the resulting function, h(x), revealed its quadratic nature and provided insights into its graphical representation. We learned how the coefficients of the quadratic function influence the parabola's shape, position, and vertex. This understanding allows us to predict the function's behavior and interpret its meaning in different contexts.

Ultimately, this exploration of function multiplication underscores the importance of a systematic approach to problem-solving. By breaking down complex problems into smaller, manageable steps, we can enhance our accuracy and deepen our understanding. This approach, coupled with a solid foundation in mathematical principles, empowers us to tackle a wide range of challenges and excel in our mathematical endeavors.