Derivative And Classification Of F(x) = (3x - 2)/(x + 2) A Comprehensive Guide

In the realm of calculus, understanding the behavior of functions is paramount. This involves delving into concepts such as derivatives, which provide insights into the rate of change of a function, and classification, which helps us categorize functions based on their properties. In this article, we will embark on a comprehensive exploration of the function f(x) = (3x - 2)/(x + 2), meticulously examining its derivative and classifying its nature. This journey will not only enhance our understanding of this particular function but also equip us with the tools and knowledge to analyze a broader range of functions encountered in mathematics and its applications. The derivative, a fundamental concept in calculus, quantifies the instantaneous rate of change of a function at a specific point. It essentially provides the slope of the line tangent to the function's graph at that point. Derivatives are instrumental in optimization problems, where we seek to find the maximum or minimum values of a function, as well as in understanding the behavior of functions, such as where they are increasing, decreasing, or have stationary points. The classification of functions, on the other hand, involves categorizing them based on their inherent properties. This can include identifying whether a function is linear, quadratic, exponential, or rational, among others. Understanding the classification of a function allows us to anticipate its behavior, predict its graph, and apply appropriate techniques for analysis. For instance, rational functions, like the one we will be examining, often exhibit unique characteristics such as vertical and horizontal asymptotes, which significantly influence their behavior.

The derivative of a function, a cornerstone of calculus, reveals the instantaneous rate of change of that function at any given point. In simpler terms, it tells us how much the function's output changes in response to a tiny change in its input. To embark on our exploration of f(x) = (3x - 2)/(x + 2), our first crucial step is to determine its derivative, denoted as f'(x). This process will unveil valuable insights into the function's behavior, such as its increasing and decreasing intervals, its critical points, and its concavity. To find the derivative of this rational function, we will employ the quotient rule, a fundamental tool in calculus specifically designed for differentiating functions that are expressed as the ratio of two other functions. The quotient rule provides a systematic approach to handling such derivatives, ensuring we accurately capture the rate of change of the overall function. The quotient rule states that if we have a function h(x) that is the quotient of two other functions, u(x) and v(x), such that h(x) = u(x) / v(x), then the derivative of h(x) is given by: h'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]². In our case, we can identify u(x) = 3x - 2 and v(x) = x + 2. Therefore, to apply the quotient rule, we first need to find the derivatives of u(x) and v(x) individually. The derivative of u(x) = 3x - 2 is simply u'(x) = 3, as the derivative of 3x is 3 and the derivative of the constant -2 is 0. Similarly, the derivative of v(x) = x + 2 is v'(x) = 1, as the derivative of x is 1 and the derivative of the constant 2 is 0. Now that we have u(x), v(x), u'(x), and v'(x), we can plug them into the quotient rule formula to find the derivative of f(x). Substituting the values, we get: f'(x) = [(3)(x + 2) - (3x - 2)(1)] / (x + 2)². Simplifying the numerator, we have: f'(x) = [3x + 6 - 3x + 2] / (x + 2)². Further simplification yields: f'(x) = 8 / (x + 2)². This is the derivative of our function f(x). This result is crucial as it will be the foundation for our subsequent analysis of the function's behavior, including determining where it is increasing or decreasing and identifying any critical points.

Now that we have successfully determined the derivative of our function, f'(x) = 8 / (x + 2)², the next crucial step is to delve into its analysis. This analysis will serve as a compass, guiding us to understand the behavior of the original function, f(x) = (3x - 2) / (x + 2). By scrutinizing the derivative, we can unearth vital information about the function's increasing and decreasing intervals, its critical points (where the function's slope is either zero or undefined), and its local extrema (maximum and minimum values). Our primary focus in this analysis is to determine the intervals where the function is either increasing or decreasing. A fundamental principle of calculus dictates that a function is increasing where its derivative is positive and decreasing where its derivative is negative. Therefore, to identify these intervals, we need to examine the sign of f'(x) across its domain. The derivative, f'(x) = 8 / (x + 2)², is a rational function, and its sign is determined by the sign of its numerator and denominator. The numerator, 8, is a positive constant, meaning it will always be positive. The denominator, (x + 2)², is a squared term, which means it will always be non-negative. However, it is crucial to note that the denominator can be zero when x = -2. This point is of particular significance as it represents a potential point of discontinuity or a critical point of the function. Since the numerator is always positive and the denominator is always non-negative, the derivative f'(x) will always be positive, except at x = -2 where it is undefined. This crucial observation tells us that the function f(x) is increasing across its entire domain, except at the point x = -2. The point x = -2 is a critical point because the derivative is undefined at this point. This is due to the denominator of the derivative becoming zero, which leads to an undefined expression. However, since the derivative does not change sign at this point (it remains positive on both sides of x = -2), it does not correspond to a local maximum or minimum. This implies that the function does not have any local extrema. In summary, by analyzing the derivative f'(x), we have gleaned valuable insights into the behavior of the original function f(x). We have determined that f(x) is increasing across its entire domain, except at x = -2 where it is undefined, and that it does not have any local extrema. These findings provide a solid foundation for our further exploration of the function, including its classification and graphical representation.

Having explored the derivative of our function, f(x) = (3x - 2) / (x + 2), and gained insights into its increasing and decreasing behavior, we now turn our attention to classifying the function itself. Classifying a function involves identifying its type or category based on its structural form and properties. This classification is crucial as it allows us to anticipate the function's behavior, predict its graph, and apply appropriate techniques for further analysis. A glimpse at the function f(x) = (3x - 2) / (x + 2) immediately reveals its defining characteristic: it is expressed as the ratio of two polynomials. The numerator, 3x - 2, is a linear polynomial, and the denominator, x + 2, is also a linear polynomial. This form is the hallmark of a rational function. A rational function is generally defined as a function that can be expressed as the quotient of two polynomials, where the denominator is not equal to zero. This broad category encompasses a wide variety of functions, each with its own unique properties and behaviors. Rational functions often exhibit intriguing characteristics such as vertical and horizontal asymptotes, which significantly influence their graphical representation and behavior. Vertical asymptotes occur at values of x where the denominator of the rational function becomes zero, leading to an undefined value for the function. In our case, the denominator x + 2 becomes zero when x = -2. This indicates that there is a vertical asymptote at x = -2. As the value of x approaches -2 from either side, the function's value will tend towards positive or negative infinity. Horizontal asymptotes, on the other hand, describe the behavior of the function as x approaches positive or negative infinity. To determine the horizontal asymptote of our function, we compare the degrees of the polynomials in the numerator and denominator. The degree of a polynomial is the highest power of the variable in the expression. In our function, both the numerator and denominator are linear polynomials, meaning they both have a degree of 1. When the degrees of the numerator and denominator are equal, the horizontal asymptote is found by dividing the leading coefficients of the polynomials. The leading coefficient is the coefficient of the term with the highest power of the variable. In our case, the leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = 3 / 1 = 3. This means that as x approaches positive or negative infinity, the function's value will approach 3. In addition to identifying the asymptotes, understanding that f(x) is a rational function also allows us to predict its overall shape and behavior. Rational functions can have a variety of shapes, including curves, hyperbolas, and combinations thereof. The presence of asymptotes often dictates the general form of the graph, with the function approaching the asymptotes but never actually crossing them. In conclusion, classifying f(x) = (3x - 2) / (x + 2) as a rational function provides us with a wealth of information about its properties and behavior. We have identified the presence of a vertical asymptote at x = -2 and a horizontal asymptote at y = 3. This classification lays the groundwork for a more comprehensive understanding of the function's graph, its domain and range, and its overall behavior.

The culmination of our analysis of f(x) = (3x - 2) / (x + 2), including finding its derivative and classifying it as a rational function, leads us to the crucial step of graphing the function. Graphing a function provides a visual representation of its behavior, allowing us to solidify our understanding and gain further insights. The graph of a function is a powerful tool for visualizing its key features, such as its domain and range, its intercepts, its asymptotes, and its increasing and decreasing intervals. By plotting the graph of f(x), we can visually confirm our previous findings and identify any additional nuances in its behavior. Before we start plotting points, let's consolidate the information we have gathered so far. We know that f(x) is a rational function with a vertical asymptote at x = -2 and a horizontal asymptote at y = 3. We also know that the function is increasing across its entire domain, except at x = -2, and that it does not have any local extrema. These pieces of information will serve as guideposts as we construct the graph. The vertical asymptote at x = -2 indicates that the function will approach positive or negative infinity as x gets closer to -2. The horizontal asymptote at y = 3 indicates that the function will approach 3 as x approaches positive or negative infinity. To accurately plot the graph, we need to identify a few key points. First, let's find the intercepts. The y-intercept is the point where the graph crosses the y-axis, which occurs when x = 0. Substituting x = 0 into our function, we get: f(0) = (3(0) - 2) / (0 + 2) = -2 / 2 = -1. Therefore, the y-intercept is at the point (0, -1). The x-intercept is the point where the graph crosses the x-axis, which occurs when f(x) = 0. Setting f(x) = 0, we get: (3x - 2) / (x + 2) = 0. A fraction is equal to zero only if its numerator is zero. Therefore, we need to solve the equation 3x - 2 = 0. Adding 2 to both sides, we get 3x = 2. Dividing both sides by 3, we get x = 2/3. Therefore, the x-intercept is at the point (2/3, 0). Now that we have the asymptotes and the intercepts, we can start plotting the graph. We can start by drawing the vertical asymptote as a dashed vertical line at x = -2 and the horizontal asymptote as a dashed horizontal line at y = 3. Then, we plot the intercepts at (0, -1) and (2/3, 0). Since we know the function is increasing across its domain, we can sketch the graph by starting from the left side of the vertical asymptote and moving upwards towards the horizontal asymptote. Similarly, on the right side of the vertical asymptote, we start from the bottom and move upwards towards the horizontal asymptote. The resulting graph will be a hyperbola-like shape, with two distinct branches separated by the vertical asymptote. One branch will approach the vertical asymptote from the left and the horizontal asymptote from below, while the other branch will approach the vertical asymptote from the right and the horizontal asymptote from above. By carefully plotting the asymptotes, intercepts, and considering the increasing nature of the function, we can accurately sketch the graph of f(x) = (3x - 2) / (x + 2). This graph provides a visual confirmation of our previous analysis and offers a comprehensive understanding of the function's behavior.

In this comprehensive exploration, we have meticulously examined the function f(x) = (3x - 2) / (x + 2), delving into its derivative, classification, and graphical representation. Our journey began with the determination of the function's derivative, f'(x) = 8 / (x + 2)², a crucial step that unveiled the function's rate of change. This derivative served as a powerful tool, allowing us to identify the intervals where the function is increasing and to pinpoint any critical points. Through careful analysis of the derivative, we discovered that f(x) is increasing across its entire domain, with the exception of x = -2 where it is undefined. We also concluded that the function does not possess any local extrema. This understanding of the function's increasing and decreasing behavior provided valuable insights into its overall characteristics. Next, we turned our attention to the classification of the function. Recognizing that f(x) is expressed as the ratio of two polynomials, we classified it as a rational function. This classification proved to be highly informative, as it allowed us to anticipate the function's behavior and predict its key features. Rational functions often exhibit unique characteristics such as vertical and horizontal asymptotes, and our analysis confirmed the presence of a vertical asymptote at x = -2 and a horizontal asymptote at y = 3. These asymptotes play a significant role in shaping the function's graph and influencing its behavior as x approaches specific values or infinity. Finally, we culminated our analysis by graphing the function. The graph provided a visual representation of our findings, solidifying our understanding of the function's behavior. By plotting the asymptotes, intercepts, and considering the increasing nature of the function, we were able to sketch an accurate representation of the function's graph. This graph showcased the characteristic shape of a rational function, with two distinct branches separated by the vertical asymptote and approaching the horizontal asymptote as x tends towards infinity. In conclusion, our comprehensive analysis of f(x) = (3x - 2) / (x + 2) has provided a thorough understanding of its derivative, classification, and graphical representation. We have successfully determined the function's derivative, classified it as a rational function, and sketched its graph, gaining valuable insights into its behavior and characteristics. This journey exemplifies the power of calculus in unraveling the intricacies of functions and provides a solid foundation for further exploration of mathematical concepts.