Analyzing G(x) = -log₄(x-3) + 2 Domain, Range, Intercepts, And Asymptotes

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In the realm of mathematical functions, logarithmic functions hold a special place, allowing us to explore relationships between numbers in a unique and insightful way. Today, we embark on a journey to dissect and understand the intricacies of the logarithmic function g(x) = -log₄(x-3) + 2. This comprehensive analysis will delve into graphing the function, determining its domain, range, and x-intercepts, and exploring its vertical and horizontal asymptotes. By the end of this exploration, you'll have a solid grasp of this function's behavior and its graphical representation.

Part A: Graphing the Logarithmic Function g(x) and Determining its Domain, Range, and x-intercepts

Graphing the Function

To effectively graph the logarithmic function g(x) = -log₄(x-3) + 2, we'll employ a step-by-step approach, carefully considering the transformations applied to the basic logarithmic function. Let's break down the components of the function:

  1. The Base Function: The foundation of our function is the logarithmic function with base 4, denoted as log₄(x). This function exhibits a characteristic logarithmic shape, increasing gradually as x increases.

  2. Horizontal Shift: The term (x-3) inside the logarithm introduces a horizontal shift. This transformation shifts the graph of log₄(x) three units to the right. As a result, the vertical asymptote, which initially resided at x = 0, now moves to x = 3.

  3. Reflection: The negative sign in front of the logarithm, -log₄(x-3), reflects the graph across the x-axis. This reflection flips the logarithmic curve, causing it to decrease as x increases.

  4. Vertical Shift: The addition of 2, -log₄(x-3) + 2, shifts the entire graph upwards by two units. This vertical translation raises the horizontal asymptote from y = 0 to y = 2.

By meticulously applying these transformations, we can accurately sketch the graph of g(x) = -log₄(x-3) + 2. The graph will exhibit a logarithmic shape, approaching the vertical asymptote at x = 3 and extending downwards as x increases. The horizontal asymptote will be located at y = 2.

Domain

The domain of a logarithmic function is restricted by the fact that the argument of the logarithm must be strictly positive. In other words, we cannot take the logarithm of zero or a negative number. For the function g(x) = -log₄(x-3) + 2, the argument of the logarithm is (x-3). Therefore, we must have:

x - 3 > 0

Solving this inequality, we find:

x > 3

Thus, the domain of g(x) is all real numbers greater than 3, which can be expressed in interval notation as (3, ∞). This means the function is defined for all values of x to the right of the vertical asymptote at x = 3.

Range

The range of a logarithmic function, on the other hand, encompasses all real numbers. As x approaches the vertical asymptote from the right, the function approaches negative infinity. As x increases, the function gradually decreases, but it never reaches a lower bound. Therefore, the range of g(x) = -log₄(x-3) + 2 is all real numbers, which can be expressed in interval notation as (-∞, ∞).

x-intercepts

The x-intercepts of a function are the points where the graph intersects the x-axis. At these points, the value of the function, g(x), is equal to zero. To find the x-intercepts of g(x) = -log₄(x-3) + 2, we set the function equal to zero and solve for x:

-log₄(x-3) + 2 = 0

Rearranging the equation, we get:

log₄(x-3) = 2

To eliminate the logarithm, we can rewrite the equation in exponential form:

4² = x - 3

Simplifying, we have:

16 = x - 3

Adding 3 to both sides, we find:

x = 19

Therefore, the x-intercept of g(x) = -log₄(x-3) + 2 is the point (19, 0). This is the only point where the graph intersects the x-axis.

Part B: Determining the Vertical and Horizontal Asymptotes

Asymptotes are lines that a function's graph approaches but never quite touches. They provide valuable information about the function's behavior as x approaches certain values or infinity. Logarithmic functions, in particular, exhibit both vertical and horizontal asymptotes.

Vertical Asymptotes

Vertical asymptotes occur where the function's value approaches infinity (or negative infinity) as x approaches a specific value. For logarithmic functions, vertical asymptotes typically arise at the boundary of the domain, where the argument of the logarithm approaches zero. In the case of g(x) = -log₄(x-3) + 2, the argument of the logarithm is (x-3). As we determined earlier, the domain of this function is x > 3. As x approaches 3 from the right, (x-3) approaches 0, and the logarithm log₄(x-3) approaches negative infinity. Consequently, the term -log₄(x-3) approaches positive infinity, and the entire function g(x) approaches positive infinity. Therefore, the vertical asymptote of g(x) = -log₄(x-3) + 2 is the vertical line x = 3.

Horizontal Asymptotes

Horizontal asymptotes describe the function's behavior as x approaches positive or negative infinity. Unlike exponential functions, which have horizontal asymptotes, basic logarithmic functions do not have horizontal asymptotes. However, in our function g(x) = -log₄(x-3) + 2, the vertical shift of +2 introduces a horizontal asymptote. As x increases without bound, the term -log₄(x-3) decreases without bound. However, the addition of 2 shifts the entire graph upwards, causing the function to approach the horizontal line y = 2. Therefore, the horizontal asymptote of g(x) = -log₄(x-3) + 2 is the horizontal line y = 2.

In conclusion, we have thoroughly analyzed the logarithmic function g(x) = -log₄(x-3) + 2. We successfully graphed the function by understanding the transformations applied to the basic logarithmic function. We determined the domain to be (3, ∞), the range to be (-∞, ∞), and the x-intercept to be (19, 0). Furthermore, we identified the vertical asymptote as x = 3 and the horizontal asymptote as y = 2. This comprehensive exploration provides a solid foundation for understanding the behavior and characteristics of logarithmic functions.

Asymptotes are crucial for understanding the behavior of functions, especially logarithmic functions. They act as guideposts, indicating how the function behaves as the input (x) approaches certain values or infinity. In this section, we will delve deeper into identifying and interpreting vertical and horizontal asymptotes, focusing on their significance in the context of logarithmic functions.

Delving Deeper into Asymptotes: Unveiling the Behavior of g(x) = -log₄(x-3) + 2

As mentioned earlier, asymptotes are lines that a function's graph approaches but never actually intersects. They are essential tools for sketching graphs and understanding the function's behavior at extreme values of x. Logarithmic functions exhibit two main types of asymptotes: vertical and horizontal.

Vertical Asymptotes: Where the Function Approaches Infinity

Vertical asymptotes occur when the function's value increases or decreases without bound as x approaches a specific value. For logarithmic functions, vertical asymptotes are intimately connected to the domain. Logarithms are only defined for positive arguments, meaning the expression inside the logarithm must be greater than zero. This restriction gives rise to vertical asymptotes.

Consider our function, g(x) = -log₄(x-3) + 2. The argument of the logarithm is (x-3). To ensure the logarithm is defined, we need:

x - 3 > 0

Solving for x, we get:

x > 3

This inequality tells us that the domain of g(x) is all real numbers greater than 3. The value x = 3 marks the boundary of the domain. As x approaches 3 from the right (values slightly greater than 3), the expression (x-3) approaches zero. The logarithm of a number approaching zero approaches negative infinity. The negative sign in front of the logarithm, -log₄(x-3), then transforms this negative infinity into positive infinity. Therefore, as x approaches 3 from the right, g(x) approaches positive infinity, indicating a vertical asymptote at x = 3.

Key Takeaway: Vertical asymptotes of logarithmic functions occur at the values of x that make the argument of the logarithm equal to zero.

Horizontal Asymptotes: The Long-Term Trend of the Function

Horizontal asymptotes describe the function's behavior as x approaches positive or negative infinity. They reveal the function's long-term trend, indicating whether the function levels off to a specific value or continues to increase or decrease indefinitely.

Basic logarithmic functions, such as log₄(x), do not have horizontal asymptotes. As x approaches infinity, the logarithm also increases without bound, albeit at a slower rate. However, transformations applied to the logarithmic function can introduce horizontal asymptotes.

In our function, g(x) = -log₄(x-3) + 2, the vertical shift of +2 plays a crucial role in creating a horizontal asymptote. As x approaches infinity, the term (x-3) also approaches infinity. The logarithm log₄(x-3) increases without bound, but the negative sign in front, -log₄(x-3), causes this term to decrease without bound (approaching negative infinity). However, the addition of 2 shifts the entire graph upwards. While -log₄(x-3) is becoming increasingly negative, the +2 provides a lower bound. The function will approach the horizontal line y = 2 as x becomes very large.

Therefore, the horizontal asymptote of g(x) = -log₄(x-3) + 2 is the line y = 2.

Key Takeaway: Horizontal asymptotes of transformed logarithmic functions are often determined by vertical shifts applied to the basic logarithmic function.

Significance of Asymptotes in Graphing

Asymptotes are invaluable aids in sketching the graph of a function. They provide a framework for understanding the function's behavior near specific values and at infinity. When graphing g(x) = -log₄(x-3) + 2, knowing the vertical asymptote at x = 3 and the horizontal asymptote at y = 2 helps us:

  1. Confine the Graph: The vertical asymptote restricts the graph to the right of the line x = 3, while the horizontal asymptote indicates the function's long-term trend as x increases.

  2. Determine the Shape: The asymptotes, combined with knowledge of the basic logarithmic shape and the transformations applied, allow us to accurately sketch the curve. We know the graph will approach the vertical asymptote as x approaches 3 from the right, and it will level off near the horizontal asymptote as x increases.

  3. Identify Key Features: The asymptotes, along with the x-intercept and any other significant points, provide a complete picture of the function's behavior.

In conclusion, a thorough understanding of asymptotes is essential for analyzing and graphing logarithmic functions. Vertical asymptotes arise from the domain restrictions of logarithms, while horizontal asymptotes are often introduced by vertical shifts. By carefully identifying and interpreting asymptotes, we gain valuable insights into the behavior and characteristics of these important mathematical functions.

Understanding the domain, range, and intercepts of a function is fundamental to grasping its overall behavior and graphical representation. These key characteristics provide valuable information about the function's input values, output values, and points of intersection with the coordinate axes. In this section, we will delve into the process of determining these properties for the logarithmic function g(x) = -log₄(x-3) + 2, providing a comprehensive analysis of its domain, range, and x-intercepts.

Deciphering the Domain, Range, and Intercepts of g(x)

Domain: Unveiling the Permissible Inputs

The domain of a function refers to the set of all possible input values (x) for which the function is defined. In simpler terms, it's the collection of x-values that we can