Evaluating The Function F(x) = 6/x A Step-by-Step Guide

In this comprehensive article, we will delve into the process of evaluating the function f(x) = 6/x for a set of given x values. This exercise is fundamental in understanding function behavior and is a crucial concept in mathematics. We will meticulously simplify each answer, ensuring clarity and accuracy. This exploration aims to provide a strong foundation for those new to function evaluation and offer a refresher for those familiar with the concepts. By the end of this guide, you'll have a solid grasp of how to substitute values into a function and simplify the results effectively. So, let's embark on this mathematical journey and unravel the intricacies of function evaluation.

Before we dive into the specific evaluations, let's first establish a clear understanding of what a function is. In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In simpler terms, it's a rule that takes a value (x, in our case) and transforms it into another value (f(x)). The expression f(x) = 6/x represents a function where the output is obtained by dividing 6 by the input value x. Understanding this basic principle is paramount to successfully evaluating functions. This foundational knowledge will enable us to approach the specific evaluations with confidence and precision. This article section serves as a cornerstone for the subsequent detailed evaluations, ensuring a coherent and progressive learning experience for the readers.

Now, we will systematically evaluate the function for each given value of x. We'll break down each step, explaining the substitution process and the simplification techniques involved. This methodical approach ensures that the underlying concepts are clearly understood and can be applied to similar problems. The primary goal is not just to arrive at the correct answers but to foster a deep understanding of how those answers are obtained. Each evaluation will be accompanied by a detailed explanation, highlighting the crucial steps and reasoning behind them. This comprehensive approach will equip the readers with the necessary tools to tackle function evaluations with confidence and accuracy.

(a) f(-5)

To evaluate f(-5), we substitute x with -5 in the function f(x) = 6/x. This gives us f(-5) = 6/(-5). Simplifying this fraction, we get f(-5) = -6/5. This straightforward substitution demonstrates the core principle of function evaluation: replacing the variable with the given value. The resulting negative fraction is a direct consequence of dividing a positive number by a negative number. This simple example sets the stage for the more complex evaluations that follow, reinforcing the fundamental concept of substituting values into a function.

(b) f(-4)

Similarly, to evaluate f(-4), we substitute x with -4 in the function f(x) = 6/x. This results in f(-4) = 6/(-4). We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2. This simplification yields f(-4) = -3/2. This example further reinforces the concept of substituting values and simplifying the resulting expressions. The simplification step highlights the importance of reducing fractions to their simplest form, a crucial skill in mathematical problem-solving. This systematic approach to function evaluation lays a strong foundation for tackling more complex functions and scenarios.

(c) f(-1/6)

Evaluating f(-1/6) involves substituting x with -1/6 in the function f(x) = 6/x. This gives us f(-1/6) = 6/(-1/6). Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite this as f(-1/6) = 6 * (-6), which simplifies to f(-1/6) = -36. This evaluation introduces the concept of dividing by fractions and the subsequent multiplication by the reciprocal. It demonstrates how seemingly complex expressions can be simplified through the application of basic mathematical principles. The result, a whole number, underscores the versatility of functions and their ability to produce diverse outputs.

(d) f(1/6)

For f(1/6), we substitute x with 1/6 in the function f(x) = 6/x. This gives us f(1/6) = 6/(1/6). Again, dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, we can rewrite this as f(1/6) = 6 * 6, which simplifies to f(1/6) = 36. This evaluation mirrors the previous one but with a positive input, highlighting the impact of the sign on the output. The resulting positive whole number reinforces the concept of reciprocals and their role in simplifying expressions involving division by fractions. This consistent application of mathematical principles across different scenarios solidifies the understanding of function evaluation.

(e) f(-1)

Finally, to evaluate f(-1), we substitute x with -1 in the function f(x) = 6/x. This gives us f(-1) = 6/(-1). Simplifying this, we get f(-1) = -6. This straightforward evaluation demonstrates the impact of dividing by -1, which simply changes the sign of the numerator. This example serves as a concise reminder of the basic principles of function evaluation and the effect of negative values on the output. The resulting negative whole number reinforces the understanding of how functions transform input values into corresponding outputs.

In conclusion, we have successfully evaluated the function f(x) = 6/x for the given values of x. Through this process, we have reinforced the fundamental concept of function evaluation: substituting the given value into the function and simplifying the resulting expression. Each evaluation has highlighted different aspects of function behavior, from the impact of negative values to the use of reciprocals when dividing by fractions. The ability to evaluate functions is a crucial skill in mathematics, serving as a building block for more advanced concepts. By mastering this skill, you'll be well-equipped to tackle a wide range of mathematical problems and applications. The step-by-step approach outlined in this article provides a solid foundation for understanding and applying function evaluation techniques.