Exploring The Function F(x) = X / (3x + 1) Domain And Evaluation

In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding relationships between variables. A function, in its essence, establishes a unique correspondence between an input value and an output value. This article delves into the intricacies of a specific function, f(x) = x / (3x + 1), exploring its properties, evaluating its values at particular points, and determining its domain. Our goal is to provide a comprehensive understanding of this function and its behavior.

To initiate our exploration, let's evaluate the function f(x) at the point x = 0.5. This involves substituting 0.5 for x in the function's expression. The calculation unfolds as follows:

f(0.5) = 0. 5 / (3 * 0.5 + 1) = 0.5 / (1.5 + 1) = 0.5 / 2.5 = 1 / 5

Therefore, the value of the function f(x) at x = 0.5 is 1/5. This calculation demonstrates the direct application of the function's definition to determine the output for a given input. Understanding this evaluation process is crucial for comprehending the function's behavior and its mapping of input values to output values. Furthermore, this simple calculation serves as a foundation for more complex analyses, such as identifying the function's range and sketching its graph. By grasping the fundamental concept of function evaluation, we can unlock deeper insights into the function's characteristics and its role in mathematical modeling.

Next, we delve into a more intricate aspect of function evaluation: finding ff(-1). This notation signifies the composition of the function with itself, meaning we first evaluate f(-1) and then use that result as the input for f(x) again. Let's break down this process step by step.

First, we evaluate f(-1):

f(-1) = -1 / (3 * -1 + 1) = -1 / (-3 + 1) = -1 / -2 = 1 / 2

Now, we use this result (1/2) as the input for f(x):

f(1/2) = (1/2) / (3 * (1/2) + 1) = (1/2) / (3/2 + 1) = (1/2) / (5/2) = 1 / 5

Thus, ff(-1) equals 1/5. This exercise illuminates the concept of function composition, a powerful tool in mathematics that allows us to combine functions and create more complex relationships. Understanding function composition is essential for modeling systems where the output of one process serves as the input for another. For example, in physics, we might use function composition to model the trajectory of a projectile, where the initial velocity affects the distance traveled, and the distance traveled, in turn, affects the final impact point. By mastering function composition, we can tackle intricate problems and gain a deeper understanding of interconnected systems.

The domain of a function encompasses all possible input values for which the function produces a valid output. In the case of f(x) = x / (3x + 1), we need to identify any values of x that would lead to an undefined result. A crucial aspect to consider is division by zero, which is mathematically undefined. Therefore, we need to find the value(s) of x that make the denominator, 3x + 1, equal to zero. Setting the denominator to zero and solving for x, we get:

3x + 1 = 0

3x = -1

x = -1 / 3

Thus, x = -1/3 is the value that cannot be included in the domain of f. This is because substituting -1/3 for x would result in division by zero, rendering the function undefined at that point. The domain of f(x) therefore consists of all real numbers except for -1/3. We can express this mathematically as:

Domain(f) = {x ∈ ℝ | x ≠ -1/3}

Understanding the domain of a function is paramount as it defines the set of valid inputs for which the function provides meaningful outputs. Identifying values excluded from the domain is crucial for avoiding mathematical errors and ensuring the function's applicability in real-world scenarios. For instance, if this function were modeling a physical process, values outside the domain might represent conditions that are physically impossible or lead to nonsensical results. Therefore, a thorough understanding of the domain is essential for the accurate and meaningful application of functions.

Throughout this exploration, we have touched upon several key concepts in mathematics, including:

• Function evaluation: The process of substituting a specific value for the variable in a function's expression to determine the corresponding output value.
• Function composition: The process of combining two or more functions by using the output of one function as the input for another.
• Domain of a function: The set of all possible input values for which the function produces a valid output.

These concepts are fundamental to understanding the behavior and properties of functions, which are essential tools in various fields, including mathematics, physics, engineering, and computer science. Functions provide a powerful way to model relationships between variables and solve complex problems.

In summary, we have thoroughly examined the function f(x) = x / (3x + 1), covering its evaluation at specific points, function composition, and domain determination. By evaluating f(0.5) and ff(-1), we demonstrated the practical application of function evaluation and composition. Furthermore, we identified x = -1/3 as the value excluded from the domain of f(x), emphasizing the importance of understanding a function's domain for its proper application. This comprehensive analysis provides a solid foundation for further exploration of functions and their applications in various mathematical and scientific contexts.

This exploration underscores the significance of functions as fundamental mathematical tools for modeling and understanding relationships between variables. By mastering the concepts of function evaluation, composition, and domain determination, we equip ourselves with the necessary skills to tackle complex problems and gain deeper insights into the world around us. The function f(x) = x / (3x + 1) serves as a valuable example for illustrating these core concepts and their practical applications. We hope this article has provided you with a clear and comprehensive understanding of this function and its properties, and that it inspires you to further explore the fascinating world of mathematics.