Graph Of F(x) = (2x-1)/(x-1) Explained

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Determining the graph of a function like f(x) = (2x-1)/(x-1) requires a comprehensive understanding of its properties and behavior. This article will delve into the intricacies of this function, exploring its key features and providing a step-by-step guide to accurately represent it graphically. We will explore aspects such as identifying asymptotes, intercepts, and general trends in the function's behavior, equipping you with the tools necessary to confidently sketch its graph.

Unveiling the Function: f(x) = (2x-1)/(x-1)

At first glance, f(x) = (2x-1)/(x-1) might seem like a straightforward rational function. However, a closer examination reveals a wealth of information crucial for understanding its graphical representation. This section breaks down the function, highlighting key components and their implications for the graph.

Identifying Asymptotes

Asymptotes are imaginary lines that a graph approaches but never quite touches. They act as guides, shaping the overall form of the curve. Our function, f(x) = (2x-1)/(x-1), presents two types of asymptotes: vertical and horizontal.

  • Vertical Asymptote: Vertical asymptotes occur where the denominator of a rational function equals zero. In our case, the denominator is x-1. Setting it to zero, we get x-1 = 0, which solves to x = 1. This means there's a vertical asymptote at x = 1. The function will approach this line infinitely closely as x gets closer to 1, but it will never actually cross it. This is because at x=1, the function is undefined due to division by zero.
  • Horizontal Asymptote: Horizontal asymptotes describe the function's behavior as x approaches positive or negative infinity. To find the horizontal asymptote, we compare the degrees of the numerator and denominator. In f(x) = (2x-1)/(x-1), both the numerator (2x-1) and the denominator (x-1) have a degree of 1 (the highest power of x is 1). When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = 2/1 = 2. This means that as x becomes very large (positive or negative), the function's value will get closer and closer to 2.

Locating Intercepts

Intercepts are points where the graph crosses the x-axis (x-intercepts) or the y-axis (y-intercepts). These points offer valuable anchors for sketching the graph. Understanding and calculating intercepts helps to accurately plot the behavior of the function on the coordinate plane.

  • X-intercept: To find the x-intercept, we set f(x) = 0 and solve for x. This means setting the numerator equal to zero: 2x - 1 = 0. Solving for x, we get 2x = 1, and thus x = 1/2. So, the x-intercept is at the point (1/2, 0). This is the point where the graph intersects the horizontal x-axis.
  • Y-intercept: To find the y-intercept, we set x = 0 and evaluate f(0). Substituting x = 0 into the function, we get f(0) = (2(0) - 1) / (0 - 1) = (-1) / (-1) = 1. So, the y-intercept is at the point (0, 1). This is the point where the graph intersects the vertical y-axis.

Analyzing Function Behavior

Beyond asymptotes and intercepts, understanding the function's overall behavior is crucial for a complete graphical representation. We need to determine where the function is increasing, decreasing, and if there are any local maxima or minima. By understanding the function's behavior, we can accurately predict how the graph will look in different regions of the coordinate plane.

  • Increasing and Decreasing Intervals: To determine where the function is increasing or decreasing, we can analyze the sign of its derivative. However, for a rational function like this, we can also reason about the behavior around the asymptotes and intercepts. The vertical asymptote divides the graph into two regions. We know the graph approaches y = 2 as x goes to positive or negative infinity. On the left side of the vertical asymptote (x = 1), the graph starts from the y-intercept (0, 1) and approaches the vertical asymptote as x approaches 1 from the left. On the right side of the vertical asymptote, the graph approaches the asymptote as x approaches 1 from the right and also approaches the horizontal asymptote as x increases. With this information, we can deduce that the function is decreasing on both intervals defined by the vertical asymptote.
  • Local Maxima and Minima: By visually inspecting the behavior and knowing the graph is decreasing in its defined intervals, it is evident that this function has neither local maxima nor local minima. The function continuously decreases as it approaches the asymptotes, indicating no turning points within its domain.

Graphing f(x) = (2x-1)/(x-1): A Step-by-Step Approach

Now that we've analyzed the key features of f(x) = (2x-1)/(x-1), let's outline a step-by-step approach to sketching its graph accurately. This process involves plotting the asymptotes and intercepts, understanding the function's behavior in different regions, and finally, connecting the points to form the complete graph.

  1. Draw the Asymptotes: Begin by drawing the vertical asymptote at x = 1 and the horizontal asymptote at y = 2. Use dashed lines to represent these asymptotes as they are not part of the graph itself but serve as guides for its behavior.
  2. Plot the Intercepts: Plot the x-intercept at (1/2, 0) and the y-intercept at (0, 1). These points will help anchor the graph and determine its position relative to the axes.
  3. Analyze Behavior Near Asymptotes: Consider the function's behavior as x approaches the vertical asymptote (x = 1) from both the left and the right. As x approaches 1 from the left (x < 1), the function approaches negative infinity. As x approaches 1 from the right (x > 1), the function approaches positive infinity. Also, consider the function's behavior as x approaches positive and negative infinity. The function approaches the horizontal asymptote (y = 2) in both cases.
  4. Sketch the Graph: Using the asymptotes and intercepts as guides, sketch the graph. Remember that the graph will approach the asymptotes but never cross them. The graph should pass through the intercepts and follow the trends determined by the behavior analysis. This involves smoothly connecting the points and ensuring the graph approaches the asymptotes appropriately.
  5. Verify the Sketch: Once you've sketched the graph, verify its accuracy by plotting a few additional points. Choose x-values in different regions and calculate the corresponding f(x) values. These additional points will confirm the shape and position of the graph, ensuring your sketch aligns with the function's properties.

Conclusion: Visualizing the Function

By carefully analyzing f(x) = (2x-1)/(x-1), we've identified its asymptotes, intercepts, and general behavior. This comprehensive approach has allowed us to develop a clear understanding of how to graph this rational function. The step-by-step method provided empowers you to accurately represent similar functions graphically, providing a visual understanding of their mathematical properties. Remember, understanding the underlying concepts is key to successfully graphing functions and interpreting their behavior.

Are you grappling with the question of which graph represents the function f(x) = (2x-1)/(x-1)? This article provides a detailed exploration of the function, walking you through the process of identifying key features and using them to match the function to its graphical representation. We will break down the function's characteristics, including asymptotes, intercepts, and its general trend, to help you visualize and understand the correct graph. Let's embark on this journey of graphical interpretation together!

Deconstructing f(x) = (2x-1)/(x-1): Key Elements for Graphing

The function f(x) = (2x-1)/(x-1) is a rational function, and understanding these types of functions is critical for accurate graphing. The shape and position of the graph are dictated by several key elements. These include vertical asymptotes, horizontal asymptotes, and intercepts. In this section, we will carefully examine each of these components to build a solid foundation for graphing.

Identifying Vertical Asymptotes

Vertical asymptotes are crucial features of rational functions. They are vertical lines that the graph approaches but never crosses. These lines are found where the denominator of the function equals zero. For our function, f(x) = (2x-1)/(x-1), we need to find the value(s) of x that make the denominator, (x-1), equal to zero. Setting (x-1) = 0 and solving for x, we get x = 1. This means there is a vertical asymptote at x = 1. This line serves as a crucial guide for sketching the graph, indicating a point where the function's value shoots off towards infinity (positive or negative).

Determining Horizontal Asymptotes

The horizontal asymptote describes the behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptote for f(x) = (2x-1)/(x-1), we compare the degrees of the polynomials in the numerator and the denominator. The degree of a polynomial is the highest power of the variable. In this case, both the numerator (2x-1) and the denominator (x-1) have a degree of 1. When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient in the numerator is 2, and in the denominator, it's 1. Therefore, the horizontal asymptote is y = 2/1 = 2. This indicates that as x gets very large (positive or negative), the function's value will approach 2.

Pinpointing Intercepts

Intercepts are the points where the graph crosses the x-axis (x-intercepts) and the y-axis (y-intercepts). They provide critical points for accurately plotting the graph. Understanding these intercepts helps in visualizing how the graph interacts with the coordinate axes.

  • X-intercept: To find the x-intercept, we set f(x) = 0 and solve for x. For a rational function, this means setting the numerator equal to zero. So, we solve 2x - 1 = 0. Adding 1 to both sides gives 2x = 1, and dividing by 2 gives x = 1/2. Therefore, the x-intercept is at the point (1/2, 0). This is where the graph crosses the horizontal x-axis.
  • Y-intercept: To find the y-intercept, we set x = 0 and evaluate f(0). Substituting x = 0 into the function gives f(0) = (2(0) - 1) / (0 - 1) = (-1) / (-1) = 1. Thus, the y-intercept is at the point (0, 1). This is the point where the graph crosses the vertical y-axis.

Matching the Graph: A Step-by-Step Guide

With the key features identified, we can now develop a strategic approach to match the function f(x) = (2x-1)/(x-1) to its graph. This involves combining our understanding of asymptotes, intercepts, and the overall behavior of rational functions to select the correct graphical representation. Following these steps will provide a systematic way to approach graphing problems and identify the right match.

  1. Locate the Vertical Asymptote: First, look for the vertical asymptote at x = 1. This means the graph will have a vertical line at x = 1 that the function approaches but never crosses. Eliminate any graphs that do not exhibit this behavior. This is a primary visual cue to narrow down the options.
  2. Identify the Horizontal Asymptote: Next, find the horizontal asymptote at y = 2. The graph should approach this horizontal line as x goes to positive or negative infinity. Eliminate graphs that do not approach y = 2 as x moves away from the origin. The horizontal asymptote provides another essential constraint on the graph's overall shape.
  3. Verify the Intercepts: Confirm that the graph has an x-intercept at (1/2, 0) and a y-intercept at (0, 1). These points must be present on the graph for it to correctly represent the function. This step helps to ensure that the graph passes through the correct points on the coordinate plane.
  4. Analyze Function Behavior: Consider how the function behaves around the asymptotes. As x approaches 1 from the left, the function approaches negative infinity. As x approaches 1 from the right, the function approaches positive infinity. The graph should reflect this behavior, showing the function diverging to infinity near the vertical asymptote. Additionally, the graph should approach the horizontal asymptote (y = 2) from both above and below as x moves away from the origin. This analysis ensures that the graph accurately portrays the function's trend and direction.
  5. Compare with Given Options: Compare the graph you've visualized with the given options. Select the graph that matches all the key features: the vertical asymptote at x = 1, the horizontal asymptote at y = 2, the x-intercept at (1/2, 0), the y-intercept at (0, 1), and the behavior near the asymptotes. This final step consolidates all the information and helps in making an informed decision.

Conclusion: Graphing with Confidence

By systematically analyzing the function f(x) = (2x-1)/(x-1) and identifying its key characteristics, we've successfully created a roadmap for matching it to its graph. Understanding the roles of asymptotes and intercepts is crucial for visualizing the behavior of rational functions. This step-by-step guide empowers you to approach similar problems with confidence and accurately interpret graphical representations of functions. Remember, practice and attention to detail are key to mastering graph matching.