Graphing G(x) = -1 + Log₃(x - 2) A Step-by-Step Guide

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In the realm of mathematical functions, logarithmic functions hold a significant place, offering a unique perspective on the relationship between variables. Understanding how to graph and analyze these functions is crucial for various applications in science, engineering, and finance. In this comprehensive guide, we will delve into the process of graphing the logarithmic function g(x) = -1 + log₃(x - 2), exploring its key features, and determining its domain and range. This exploration will empower you to confidently analyze and interpret logarithmic functions in various contexts.

1. Understanding the Logarithmic Function

Before we dive into graphing the specific function g(x) = -1 + log₃(x - 2), let's establish a solid understanding of logarithmic functions in general. A logarithmic function is the inverse of an exponential function. The logarithmic function y = logₐ(x) (read as "log base a of x") answers the question: "To what power must we raise the base a to get x?" Here, a is the base of the logarithm, and it must be a positive number not equal to 1. The value x must be positive.

In simpler terms, the logarithm tells you the exponent needed to reach a certain number from a specific base. For example, log₁₀(100) = 2 because 10² = 100. Similarly, log₂(8) = 3 because 2³ = 8. This foundational understanding is crucial for grasping the behavior and characteristics of logarithmic functions, especially when dealing with transformations and graphing.

The logarithmic function has a vertical asymptote at x = 0. This means that the function approaches the vertical line x = 0 but never actually touches it. As x approaches 0 from the right, the value of the logarithmic function approaches negative infinity. This asymptotic behavior is a key characteristic of logarithmic functions and is essential for accurate graphing. Furthermore, logarithmic functions are only defined for positive values of x. This is because you cannot raise a positive base to any power and get a non-positive result. Understanding these limitations is critical when determining the domain of a logarithmic function.

2. Deconstructing g(x) = -1 + log₃(x - 2)

Now, let's turn our attention to the specific logarithmic function g(x) = -1 + log₃(x - 2). This function is a transformation of the basic logarithmic function y = log₃(x). Understanding these transformations is key to accurately graphing the function.

  • log₃(x): This is the basic logarithmic function with base 3. Its graph passes through the point (1, 0) and has a vertical asymptote at x = 0.
  • log₃(x - 2): This represents a horizontal shift of the basic logarithmic function 2 units to the right. The vertical asymptote shifts from x = 0 to x = 2. This is because the argument of the logarithm, (x - 2), must be positive, so x > 2.
  • -1 + log₃(x - 2): This represents a vertical shift of the function log₃(x - 2) downwards by 1 unit. This shift affects the overall position of the graph but does not change the asymptote, which remains at x = 2. By recognizing these transformations, we can systematically build the graph of the function, starting from the basic logarithmic form and applying each transformation step-by-step.

These transformations are not arbitrary; they directly impact the graph's position and shape. The horizontal shift is particularly important because it alters the function's domain and the location of the vertical asymptote. The vertical shift, on the other hand, simply moves the graph up or down without affecting the domain or asymptote. Understanding the interplay of these transformations is fundamental to accurately interpreting and manipulating logarithmic functions.

3. Plotting Key Points

To graph the function g(x) = -1 + log₃(x - 2), we need to identify at least two key points on the graph. These points will help us visualize the shape and position of the logarithmic curve. To find these points, we can choose values of x that make the argument of the logarithm (x - 2) a power of 3. This simplifies the calculation of the logarithm.

  • Point 1: Let's choose x = 3. This gives us g(3) = -1 + log₃(3 - 2) = -1 + log₃(1). Since log₃(1) = 0 (because 3⁰ = 1), we have g(3) = -1 + 0 = -1. Therefore, the point (3, -1) lies on the graph.
  • Point 2: Let's choose x = 5. This gives us g(5) = -1 + log₃(5 - 2) = -1 + log₃(3). Since log₃(3) = 1 (because 3¹ = 3), we have g(5) = -1 + 1 = 0. Therefore, the point (5, 0) lies on the graph.

By carefully selecting values of x, we can determine corresponding y-values that provide clear reference points for graphing the function. These points, combined with our understanding of the function's asymptotic behavior, will allow us to sketch an accurate representation of the logarithmic curve. It is also helpful to choose points on either side of the asymptote to better understand the function's behavior near the vertical line.

4. Drawing the Asymptote

As we discussed earlier, the function g(x) = -1 + log₃(x - 2) has a vertical asymptote. The asymptote is a vertical line that the graph of the function approaches but never intersects. To find the equation of the asymptote, we set the argument of the logarithm equal to zero and solve for x.

In this case, we have x - 2 = 0, which gives us x = 2. Therefore, the vertical asymptote is the line x = 2. This is a crucial piece of information for graphing the function, as it defines the boundary of the function's domain and influences its overall shape.

When graphing the function, it is essential to draw the asymptote as a dashed vertical line at x = 2. This visual cue helps us understand that the function gets infinitely close to this line but never crosses it. The asymptote essentially acts as a guide for sketching the logarithmic curve, ensuring that the graph accurately reflects the function's behavior near this boundary. Without accurately plotting the asymptote, it is easy to misrepresent the function's true shape and domain.

5. Sketching the Graph

Now that we have two points (3, -1) and (5, 0) and the vertical asymptote x = 2, we can sketch the graph of the function g(x) = -1 + log₃(x - 2). Start by plotting the two points on a coordinate plane. Then, draw a dashed vertical line at x = 2 to represent the asymptote. Remember that the graph will approach the asymptote as x gets closer to 2 from the right, but it will never cross it.

Since this is a logarithmic function with a base greater than 1 (base 3), the graph will increase as x increases. The graph will start close to the asymptote and gradually curve upwards, passing through the plotted points. The shape of the graph will resemble a typical logarithmic curve, but it will be shifted 2 units to the right and 1 unit downwards due to the transformations in the function.

It is also helpful to consider the behavior of the function as x becomes very large. As x approaches infinity, the value of log₃(x - 2) also approaches infinity, and therefore, g(x) approaches infinity as well. This indicates that the graph will continue to rise as x increases, albeit at a decreasing rate. By connecting the plotted points and following the asymptotic behavior, you can create an accurate sketch of the logarithmic function g(x) = -1 + log₃(x - 2).

6. Determining the Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the logarithmic function g(x) = -1 + log₃(x - 2), the argument of the logarithm (x - 2) must be positive. Therefore, we have the inequality x - 2 > 0, which gives us x > 2. This means that the domain of the function is all real numbers greater than 2. We can express this in interval notation as (2, ∞).

The range of a function is the set of all possible output values (y-values) that the function can produce. For logarithmic functions, the range is always all real numbers. This is because the logarithm can take on any real value, depending on the input. The vertical shift of -1 in the function g(x) = -1 + log₃(x - 2) does not affect the range. Therefore, the range of the function is all real numbers, which we can express in interval notation as (-∞, ∞).

Understanding the domain and range is crucial for fully characterizing a function. The domain tells us the limits of the input values, while the range tells us the extent of the output values. These concepts are fundamental for interpreting the function's behavior and its applications in various contexts.

7. Conclusion

In this comprehensive guide, we have explored the process of graphing the logarithmic function g(x) = -1 + log₃(x - 2). We began by establishing a solid understanding of logarithmic functions and their key properties. We then deconstructed the given function, identifying the transformations applied to the basic logarithmic form. By plotting key points, drawing the asymptote, and considering the function's behavior, we were able to sketch an accurate graph.

Finally, we determined the domain and range of the function, providing a complete characterization of its behavior. By mastering these techniques, you can confidently graph and analyze logarithmic functions, unlocking their potential in various mathematical and real-world applications. Understanding the nuances of logarithmic functions, such as asymptotes and transformations, is key to accurately interpreting their behavior and applying them to various problems.

This knowledge is not just limited to theoretical exercises; it has practical implications in fields such as finance, where logarithmic scales are used to represent exponential growth, and in physics, where logarithmic relationships describe phenomena like sound intensity and radioactive decay. Therefore, a solid grasp of graphing and analyzing logarithmic functions is a valuable asset in any STEM-related field.