Evaluating The Product Of Functions G(n) And H(n) At N=1

In mathematics, functions are fundamental building blocks that describe relationships between variables. Often, we encounter situations where we need to combine functions in various ways, one of which is through multiplication. This article delves into the concept of multiplying functions and provides a detailed, step-by-step approach to evaluating the product of two functions at a specific point. We will use the example provided to illustrate the process clearly.

Introduction to Function Operations

Before we dive into the specific problem, let's briefly discuss function operations. Just like numbers, functions can be added, subtracted, multiplied, and divided. When we perform these operations on functions, we create new functions. For instance, if we have two functions, f(x) and g(x), we can define their sum as (f + g)(x) = f(x) + g(x), their difference as (f - g)(x) = f(x) - g(x), their product as (f ⋅ g)(x) = f(x) ⋅ g(x), and their quotient as (f / g)(x) = f(x) / g(x), provided that g(x) ≠ 0. Understanding these operations is crucial for solving various mathematical problems, especially those involving composite functions.

Deep Dive into Function Multiplication

In the realm of function operations, function multiplication holds a significant place. When we talk about the product of two functions, say f(x) and g(x), we are essentially creating a new function by multiplying the outputs of the individual functions for the same input x. Symbolically, this is represented as (f ⋅ g)(x) = f(x) ⋅ g(x). This operation is fundamental in various mathematical contexts, including calculus, algebra, and mathematical modeling. For instance, in physics, you might multiply a function representing force by a function representing distance to calculate work. In economics, you might multiply a function representing price by a function representing quantity to calculate revenue. The key to mastering function multiplication lies in understanding how to apply this definition correctly and efficiently, especially when dealing with complex function expressions. Function multiplication isn't just a mathematical abstraction; it's a tool that helps us understand and model real-world phenomena. This operation allows us to combine different relationships and behaviors into a single, cohesive model. For example, imagine modeling the growth of a population where the growth rate depends on both the current population size and the available resources. Function multiplication could be used to combine a function representing population growth with a function representing resource availability, providing a more comprehensive model of the population dynamics. In essence, mastering function multiplication is about gaining the ability to create more sophisticated and accurate mathematical models.

The Importance of Evaluating at a Specific Point

When working with functions, it's often necessary to evaluate the function at a specific point. This means substituting a particular value for the variable (usually x or n) and calculating the resulting output. Evaluating functions at specific points is essential for several reasons. First, it allows us to understand the behavior of the function at that point, such as its value, whether it's increasing or decreasing, or if it has any local extrema. Second, it's crucial for applications of functions in real-world problems. For example, if a function represents the temperature at a given time, evaluating the function at a specific time will tell us the temperature at that moment. Third, evaluating functions is a fundamental step in many mathematical procedures, such as finding limits, derivatives, and integrals. When we evaluate composite functions, the process involves first performing the function operation (e.g., addition, subtraction, multiplication, or division) and then substituting the specific value. This order of operations is critical to obtaining the correct result. Evaluating functions at a specific point transforms a general functional relationship into a concrete numerical value, providing insights and enabling further analysis. This is why it's a cornerstone skill in mathematics and its applications.

Problem Statement: Finding (g ⋅ h)(1)

We are given two functions:

• g(n) = n² + 4 + 2n
• h(n) = -3n + 2

Our goal is to find (g ⋅ h)(1), which represents the product of the functions g(n) and h(n) evaluated at n = 1. This means we need to first find the product function (g ⋅ h)(n) and then substitute n = 1 into the resulting expression. This problem exemplifies the application of function multiplication and evaluation at a specific point, reinforcing the concepts we discussed earlier.

Step 1: Find the Product Function (g ⋅ h)(n)

The first step is to determine the product function (g ⋅ h)(n). According to the definition of function multiplication, we have:

(g ⋅ h)(n) = g(n) ⋅ h(n)

Substitute the given expressions for g(n) and h(n):

(g ⋅ h)(n) = (n² + 4 + 2n) ⋅ (-3n + 2)

Now, we need to expand this product. We can use the distributive property (also known as the FOIL method) to multiply the two expressions:

(g ⋅ h)(n) = n²(-3n + 2) + 2n(-3n + 2) + 4(-3n + 2)

Distribute each term:

(g ⋅ h)(n) = -3n³ + 2n² - 6n² + 4n - 12n + 8

Combine like terms to simplify the expression:

(g ⋅ h)(n) = -3n³ - 4n² - 8n + 8

So, the product function (g ⋅ h)(n) is -3n³ - 4n² - 8n + 8. This expression represents the new function formed by multiplying the original functions g(n) and h(n). The process of finding this product function involves algebraic manipulation, specifically the distributive property and combining like terms. This step is crucial because it transforms the problem from evaluating two separate functions into evaluating a single, combined function. Understanding how to correctly perform this algebraic manipulation is essential for solving problems involving function operations. It lays the groundwork for the next step, which is evaluating this product function at a specific point.

Step 2: Evaluate (g ⋅ h)(1)

Now that we have found the product function (g ⋅ h)(n) = -3n³ - 4n² - 8n + 8, the next step is to evaluate this function at n = 1. This means we substitute n = 1 into the expression for (g ⋅ h)(n) and calculate the result. This process is straightforward but requires careful attention to arithmetic operations. Substituting n = 1 into the expression, we get:

(g ⋅ h)(1) = -3(1)³ - 4(1)² - 8(1) + 8

Now, we simplify the expression step by step. First, we calculate the powers:

(g ⋅ h)(1) = -3(1) - 4(1) - 8(1) + 8

Next, we perform the multiplications:

(g ⋅ h)(1) = -3 - 4 - 8 + 8

Finally, we perform the additions and subtractions from left to right:

(g ⋅ h)(1) = -7 - 8 + 8

(g ⋅ h)(1) = -15 + 8

(g ⋅ h)(1) = -7

Therefore, (g ⋅ h)(1) = -7. This result is the value of the product of the functions g(n) and h(n) when evaluated at n = 1. The process of evaluating the function at a specific point transforms the functional expression into a concrete numerical value. This value provides insight into the behavior of the combined function at that particular input. In this case, we found that the product of the functions g(n) and h(n) at n = 1 is -7.

Conclusion

In this article, we have demonstrated how to find the product of two functions and evaluate it at a specific point. We followed a step-by-step approach, first finding the product function (g ⋅ h)(n) and then substituting n = 1 into the resulting expression. The final answer is (g ⋅ h)(1) = -7. This process highlights the importance of understanding function operations and algebraic manipulation in solving mathematical problems. By mastering these concepts, you can tackle more complex problems involving functions and their applications.

Understanding function multiplication and evaluation is crucial for various mathematical and real-world applications. This example provides a solid foundation for further exploration of function operations and their uses in different contexts. Whether you're a student learning these concepts for the first time or someone looking to refresh your knowledge, this step-by-step guide provides a clear and concise explanation of the process. The ability to work with functions is a valuable skill in many fields, and this article aims to equip you with the tools you need to succeed.