Analyzing The Rational Function H(x) = (3x - 8) / (x + 1)

In this comprehensive exploration, we delve into the intricacies of the rational function h(x) = (3x - 8) / (x + 1). This function, a ratio of two polynomials, presents a rich landscape of mathematical properties to investigate. Our analysis will encompass several key aspects, including the function's domain, intercepts, asymptotes, and overall behavior. We will dissect each of these elements to gain a profound understanding of the function's graphical representation and its mathematical characteristics. Rational functions like this one are fundamental in various fields, such as physics, engineering, and economics, where they model real-world phenomena involving ratios and rates of change. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of h(x) = (3x - 8) / (x + 1), we need to identify any values of x that would make the denominator equal to zero, as division by zero is undefined. Setting the denominator, x + 1, equal to zero, we find that x = -1. Therefore, the domain of the function is all real numbers except for x = -1. This can be expressed in interval notation as (-∞, -1) U (-1, ∞). Understanding the domain is crucial as it defines the boundaries within which the function operates and where its graph exists. The intercepts of a function are the points where the graph of the function intersects the x-axis and the y-axis. The x-intercept(s) occur when h(x) = 0, and the y-intercept occurs when x = 0. To find the x-intercept(s), we set the numerator of the function equal to zero: 3x - 8 = 0. Solving for x, we get x = 8/3. Thus, the x-intercept is the point (8/3, 0). To find the y-intercept, we substitute x = 0 into the function: h(0) = (3(0) - 8) / (0 + 1) = -8. Therefore, the y-intercept is the point (0, -8). These intercepts provide valuable anchor points for sketching the graph of the function and understanding its behavior near the axes. The asymptotes of a rational function are lines that the graph of the function approaches but never touches or crosses. There are three types of asymptotes: vertical, horizontal, and oblique (or slant). Vertical asymptotes occur at values of x where the denominator of the function equals zero, which we have already identified as x = -1. This is because as x approaches -1, the denominator approaches zero, causing the function value to approach infinity (or negative infinity). Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptote, we compare the degrees of the numerator and denominator polynomials. In this case, both the numerator (3x - 8) and the denominator (x + 1) are linear (degree 1). When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1, so the horizontal asymptote is y = 3/1 = 3. This means that as x becomes very large (positive or negative), the function values approach 3. Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. In this case, the degrees are equal, so there is no oblique asymptote.

Let's first delve into the domain and intercepts of the rational function h(x) = (3x - 8) / (x + 1). Understanding these fundamental aspects is crucial for grasping the function's behavior and graphical representation. The domain, in mathematical terms, refers to the set of all possible input values (x-values) for which the function is defined and produces a real number output. In the context of rational functions, we must be particularly mindful of the denominator. A rational function is undefined when its denominator equals zero, as division by zero is an undefined operation in mathematics. Therefore, to determine the domain of h(x), we need to identify any x-values that would make the denominator, x + 1, equal to zero. Setting x + 1 = 0, we quickly find that x = -1. This value is excluded from the domain because it would result in division by zero. Consequently, the domain of h(x) encompasses all real numbers except for x = -1. We can express this mathematically using interval notation as (-∞, -1) U (-1, ∞), where the union symbol (U) indicates that the domain consists of two separate intervals: all numbers less than -1 and all numbers greater than -1. In essence, the function is defined for any real number input except for -1. Next, we turn our attention to the intercepts of the function. Intercepts are the points where the graph of the function intersects the coordinate axes. The x-intercept(s) are the points where the graph crosses the x-axis, which occurs when the function's value, h(x), is equal to zero. The y-intercept is the point where the graph crosses the y-axis, which occurs when the input value, x, is equal to zero. To find the x-intercept(s), we set the function equal to zero and solve for x: (3x - 8) / (x + 1) = 0. A fraction is equal to zero if and only if its numerator is equal to zero. Therefore, we need to solve the equation 3x - 8 = 0. Adding 8 to both sides gives 3x = 8, and dividing by 3 yields x = 8/3. Thus, the x-intercept is the point (8/3, 0). This point represents where the graph of the function crosses the x-axis. To find the y-intercept, we substitute x = 0 into the function: h(0) = (3(0) - 8) / (0 + 1) = -8. Therefore, the y-intercept is the point (0, -8). This point represents where the graph of the function crosses the y-axis. The intercepts provide crucial reference points for sketching the graph of the function. They help us visualize where the function intersects the axes and how it behaves in the vicinity of those points. Furthermore, the domain informs us about any restrictions on the x-values, ensuring that we avoid any undefined points in our graph. Together, the domain and intercepts lay the foundation for a comprehensive understanding of the function's graphical representation.

The asymptotes of a rational function are lines that the graph of the function approaches but never touches or crosses. Understanding asymptotes is crucial for accurately sketching the graph of the function and comprehending its behavior as x approaches certain values or infinity. There are three primary types of asymptotes: vertical, horizontal, and oblique (or slant). In this section, we will focus on vertical and horizontal asymptotes for the rational function h(x) = (3x - 8) / (x + 1). Vertical asymptotes occur at values of x where the denominator of the rational function equals zero. As we determined in the domain analysis, the denominator x + 1 is equal to zero when x = -1. This means that as x approaches -1 from either the left or the right, the denominator approaches zero, causing the function value to approach infinity (either positive or negative infinity). Consequently, there is a vertical asymptote at the line x = -1. The vertical asymptote acts as a barrier that the graph of the function cannot cross. It represents a point where the function becomes unbounded and its values shoot off towards infinity or negative infinity. The behavior of the function near the vertical asymptote is an essential characteristic that helps define the overall shape of the graph. Horizontal asymptotes, on the other hand, describe the behavior of the function as x approaches positive or negative infinity. They indicate the value that the function approaches as x becomes extremely large or extremely small. To find the horizontal asymptote, we compare the degrees of the numerator and denominator polynomials. The degree of a polynomial is the highest power of the variable (x) in the polynomial. In the case of h(x) = (3x - 8) / (x + 1), both the numerator (3x - 8) and the denominator (x + 1) are linear, meaning they have a degree of 1. When the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient is the coefficient of the term with the highest power of x. In the numerator, the leading coefficient is 3 (the coefficient of 3x), and in the denominator, the leading coefficient is 1 (the coefficient of x). Therefore, the horizontal asymptote is y = 3/1 = 3. This means that as x approaches positive or negative infinity, the function values approach 3. The graph of the function will get closer and closer to the line y = 3 but will never actually cross it (though it may cross the horizontal asymptote at other points). The horizontal asymptote provides valuable information about the long-term behavior of the function. It indicates the value that the function tends towards as x moves far away from the origin. Understanding both vertical and horizontal asymptotes is essential for sketching the graph of a rational function accurately. They define the boundaries of the graph and provide crucial information about its behavior in different regions of the coordinate plane. Vertical asymptotes highlight points where the function becomes unbounded, while horizontal asymptotes reveal the function's long-term behavior as x approaches infinity.

To gain a comprehensive understanding of the rational function h(x) = (3x - 8) / (x + 1), it is crucial to analyze its behavior in various intervals and regions of the coordinate plane. This involves examining how the function values change as x increases or decreases, identifying intervals where the function is increasing or decreasing, and determining the concavity of the graph. By carefully studying these aspects, we can develop a detailed picture of the function's graphical representation and its mathematical characteristics. We begin by considering the critical points of the function, which are the points where the derivative of the function is either zero or undefined. The derivative provides information about the slope of the tangent line to the graph of the function at any given point. A positive derivative indicates that the function is increasing, a negative derivative indicates that the function is decreasing, and a zero derivative indicates a stationary point (a local maximum or minimum). To find the derivative of h(x), we can use the quotient rule, which states that if h(x) = u(x) / v(x), then h'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2. In this case, u(x) = 3x - 8 and v(x) = x + 1. The derivatives of these functions are u'(x) = 3 and v'(x) = 1. Applying the quotient rule, we get:

h'(x) = [3(x + 1) - (3x - 8)(1)] / (x + 1)^2 = (3x + 3 - 3x + 8) / (x + 1)^2 = 11 / (x + 1)^2

Notice that the derivative h'(x) = 11 / (x + 1)^2 is always positive for all x in the domain of h(x) (i.e., all x except x = -1). This is because the numerator is a positive constant (11), and the denominator is a square, which is always non-negative. Since the derivative is always positive, the function h(x) is always increasing on its domain. This means that as x increases, the function values also increase. However, it is important to remember that the function has a vertical asymptote at x = -1, so the function is increasing on the intervals (-∞, -1) and (-1, ∞) separately. To further analyze the function's behavior, we can examine its concavity. Concavity refers to the direction in which the graph of the function is curved. A function is concave up if its graph is curved upwards, and it is concave down if its graph is curved downwards. The concavity of a function is determined by the sign of its second derivative. To find the second derivative of h(x), we need to differentiate h'(x). Using the power rule and chain rule, we get:

h''(x) = -22 / (x + 1)^3

Now, we analyze the sign of h''(x). The numerator is a negative constant (-22). The denominator (x + 1)^3 is positive when x > -1 and negative when x < -1. Therefore, h''(x) is negative when x > -1 (meaning the function is concave down) and positive when x < -1 (meaning the function is concave up). This tells us that the graph of the function is curved upwards to the left of the vertical asymptote (x = -1) and curved downwards to the right of the vertical asymptote. By combining the information about the function's increasing/decreasing behavior and its concavity, we can create a detailed sketch of its graph. The function is increasing on both sides of the vertical asymptote, concave up to the left of the asymptote, and concave down to the right of the asymptote. This provides a clear picture of how the function behaves as it approaches the asymptote and as x moves towards positive or negative infinity. In conclusion, analyzing the function's behavior involves a multi-faceted approach, including finding critical points, determining intervals of increasing and decreasing behavior, and examining concavity. By carefully studying these aspects, we can gain a deep understanding of the function's graphical representation and its mathematical characteristics.

To graph the rational function h(x) = (3x - 8) / (x + 1) effectively, we synthesize all the information gathered from our previous analyses. This includes the domain, intercepts, asymptotes, and the function's increasing/decreasing behavior and concavity. By systematically combining these elements, we can create an accurate and informative visual representation of the function. First, we establish the coordinate axes and mark the key features we have identified. The domain is all real numbers except x = -1, which means there is a vertical asymptote at x = -1. We draw a dashed vertical line at x = -1 to represent this asymptote. The function has an x-intercept at (8/3, 0) and a y-intercept at (0, -8). We plot these points on the coordinate plane. The horizontal asymptote is y = 3. We draw a dashed horizontal line at y = 3 to represent this asymptote. Next, we consider the function's increasing/decreasing behavior. We determined that the function is always increasing on its domain, meaning that as x increases, the function values also increase. However, we must remember that the function is increasing on the intervals (-∞, -1) and (-1, ∞) separately due to the vertical asymptote. We also analyzed the concavity of the function. The function is concave up when x < -1 and concave down when x > -1. This means that the graph is curved upwards to the left of the vertical asymptote and curved downwards to the right of the vertical asymptote. Now, we can begin to sketch the graph. Starting from the left side of the graph (x < -1), we know that the function is increasing and concave up. As x approaches -∞, the function approaches the horizontal asymptote y = 3. As x approaches -1 from the left, the function approaches -∞ due to the vertical asymptote. We draw a curve that follows these characteristics, starting near the horizontal asymptote at the left, curving upwards, and approaching the vertical asymptote as x gets closer to -1. On the right side of the graph (x > -1), the function is increasing and concave down. As x approaches -1 from the right, the function approaches +∞ due to the vertical asymptote. As x approaches +∞, the function approaches the horizontal asymptote y = 3. We draw a curve that follows these characteristics, starting near the vertical asymptote on the right, curving downwards, and approaching the horizontal asymptote as x gets larger. We also make sure that the graph passes through the x-intercept (8/3, 0) and y-intercept (0, -8). The completed graph should show two distinct curves, one on each side of the vertical asymptote. The left curve is increasing and concave up, approaching the vertical asymptote as x approaches -1 from the left and approaching the horizontal asymptote as x approaches -∞. The right curve is increasing and concave down, approaching the vertical asymptote as x approaches -1 from the right and approaching the horizontal asymptote as x approaches +∞. The intercepts provide additional anchor points for the graph, ensuring that it accurately represents the function's behavior near the axes. Graphing a rational function is a process of combining information from various analyses to create a visual representation of the function's characteristics. By carefully considering the domain, intercepts, asymptotes, increasing/decreasing behavior, and concavity, we can produce an accurate and informative graph that reveals the function's overall shape and behavior.

In conclusion, our exploration of the rational function h(x) = (3x - 8) / (x + 1) has provided a comprehensive understanding of its mathematical properties and graphical representation. We have dissected the function into its fundamental components, analyzing its domain, intercepts, asymptotes, and behavior to gain a holistic perspective. Our journey began with identifying the domain, which revealed the restriction x ≠ -1 due to the denominator. This led us to the discovery of a vertical asymptote at x = -1, a crucial feature that dictates the function's behavior near this point. We then determined the intercepts, finding the x-intercept at (8/3, 0) and the y-intercept at (0, -8). These intercepts serve as anchor points for sketching the graph and understanding its relationship with the coordinate axes. The analysis of asymptotes extended to the horizontal asymptote at y = 3, which describes the function's long-term behavior as x approaches positive or negative infinity. Understanding the horizontal asymptote is essential for grasping the function's trend as it moves away from the origin. By examining the function's derivative, we established that it is always increasing on its domain. This characteristic provides valuable insight into the function's overall trend and how its values change as x increases or decreases. The second derivative analysis revealed the concavity of the function, indicating that it is concave up when x < -1 and concave down when x > -1. This information is crucial for accurately depicting the curvature of the graph on either side of the vertical asymptote. Finally, we synthesized all these findings to create a detailed graph of the function. The graph showcases the vertical asymptote, the horizontal asymptote, the intercepts, and the increasing behavior and concavity. It provides a visual representation of the function's characteristics and allows us to observe its behavior in different regions of the coordinate plane. The process of analyzing a rational function like h(x) = (3x - 8) / (x + 1) exemplifies the power of mathematical tools and techniques in understanding complex relationships. By systematically dissecting the function and examining its various aspects, we can develop a deep appreciation for its mathematical properties and its graphical representation. Rational functions play a significant role in various fields, including physics, engineering, and economics, where they model real-world phenomena involving ratios and rates of change. The skills and knowledge gained from analyzing this function can be applied to a wide range of mathematical problems and real-world applications. This comprehensive exploration serves as a testament to the beauty and utility of mathematics in unraveling the complexities of functions and their behavior. The ability to analyze and graph rational functions is a valuable asset for anyone pursuing further studies in mathematics, science, or engineering. It provides a foundation for understanding more advanced concepts and solving real-world problems that involve mathematical modeling. The insights gained from this analysis will undoubtedly serve as a valuable resource for future mathematical endeavors.