Evaluating (f G)(0) For F(x) = 3 - 2x And G(x) = 1/(x+5)

In the realm of mathematics, composite functions play a crucial role in understanding the behavior and relationships between different mathematical expressions. Among these operations, the product of functions, denoted as (f ⋅ g)(x), is a fundamental concept. This article delves into the process of evaluating the product of two functions, f(x) and g(x), at a specific point, in this case, x = 0. We will explore the step-by-step method to determine the value of (f ⋅ g)(0) when f(x) = 3 - 2x and g(x) = 1/(x+5). This exploration will not only enhance your understanding of function operations but also provide a practical application of these concepts. Let's embark on this mathematical journey to unravel the intricacies of function products and their evaluation.

Defining the Functions: f(x) and g(x)

Before we delve into the evaluation of (f ⋅ g)(0), it is imperative to define the functions f(x) and g(x) explicitly. These functions serve as the building blocks for our subsequent calculations. Understanding their individual properties and behavior is essential for grasping the concept of their product. Let's take a closer look at each function:

• Function f(x): The function f(x) is defined as a linear function, expressed as f(x) = 3 - 2x. This function represents a straight line when plotted on a graph, with a slope of -2 and a y-intercept of 3. The slope indicates the rate at which the function's value changes with respect to x, while the y-intercept represents the function's value when x is equal to 0. Linear functions are fundamental in mathematics and have wide-ranging applications in various fields.
• Function g(x): The function g(x) is defined as a rational function, expressed as g(x) = 1/(x+5). This function represents a hyperbola when plotted on a graph, with a vertical asymptote at x = -5. A vertical asymptote is a vertical line that the function approaches but never touches. In this case, the function approaches the line x = -5 as x gets closer to -5, but it never actually intersects the line. Rational functions are characterized by having a variable in the denominator and can exhibit unique behaviors, such as asymptotes and discontinuities.

Understanding the individual characteristics of f(x) and g(x) is crucial for comprehending how they interact when their product is evaluated. The linear nature of f(x) and the rational nature of g(x) will influence the behavior of their product, (f ⋅ g)(x).

Understanding the Product of Functions: (f ⋅ g)(x)

The product of two functions, denoted as (f ⋅ g)(x), represents a new function formed by multiplying the outputs of the individual functions f(x) and g(x) for the same input value x. This operation combines the behaviors of the individual functions, resulting in a new function with potentially unique characteristics. To understand the product of functions, let's delve into its definition and properties:

• Definition: The product of functions (f ⋅ g)(x) is defined as (f ⋅ g)(x) = f(x) * g(x). This means that for any given value of x, we first evaluate f(x) and g(x) separately, and then multiply the results to obtain the value of (f ⋅ g)(x).
• Properties: The product of functions inherits properties from both f(x) and g(x). For instance, if either f(x) or g(x) is continuous at a point, then (f ⋅ g)(x) is also continuous at that point. Similarly, if either f(x) or g(x) is differentiable at a point, then (f ⋅ g)(x) is also differentiable at that point. However, the specific behavior of (f ⋅ g)(x) will depend on the individual characteristics of f(x) and g(x).

In our case, f(x) = 3 - 2x and g(x) = 1/(x+5). Therefore, the product of these functions is given by:

(f ⋅ g)(x) = f(x) * g(x) = (3 - 2x) * (1/(x+5))

This expression represents the product of the linear function f(x) and the rational function g(x). To evaluate (f ⋅ g)(0), we need to substitute x = 0 into this expression.

Evaluating (f ⋅ g)(0)

Now that we have defined the functions f(x) and g(x) and understood the concept of their product, we can proceed to evaluate (f ⋅ g)(0). This involves substituting x = 0 into the expression for (f ⋅ g)(x) and simplifying the result. Let's break down the process step by step:

1. Substitute x = 0 into the expression for (f ⋅ g)(x):

   We have (f ⋅ g)(x) = (3 - 2x) * (1/(x+5)). Substituting x = 0, we get:

   (f ⋅ g)(0) = (3 - 2(0)) * (1/(0+5))

2. Simplify the expression:

   Now, we simplify the expression by performing the arithmetic operations:

   (f ⋅ g)(0) = (3 - 0) * (1/5)

   (f ⋅ g)(0) = 3 * (1/5)

   (f ⋅ g)(0) = 3/5

Therefore, the value of (f ⋅ g)(0) is 3/5. This means that when x is equal to 0, the product of the functions f(x) and g(x) is equal to 3/5. This result provides a specific point on the graph of the function (f ⋅ g)(x).

Conclusion

In conclusion, we have successfully evaluated the value of (f ⋅ g)(0) given the functions f(x) = 3 - 2x and g(x) = 1/(x+5). By understanding the definitions of the functions, the concept of their product, and the step-by-step evaluation process, we have arrived at the result (f ⋅ g)(0) = 3/5. This exercise demonstrates the importance of function operations in mathematics and provides a practical application of these concepts. The product of functions allows us to combine the behaviors of individual functions, creating new functions with unique characteristics. Evaluating these functions at specific points provides valuable information about their behavior and properties. Understanding these concepts is crucial for further exploration in mathematics and its applications.

This exploration of function products and their evaluation serves as a foundation for more advanced topics in mathematics, such as calculus and analysis. The ability to manipulate and evaluate functions is essential for solving complex problems and understanding mathematical relationships. By mastering these fundamental concepts, we can unlock the power of mathematics to describe and explain the world around us.