In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log2 (x-2)

Verified step by step guidance

1. Step 1: Start by understanding the base function f(x) = log₂(x). This is the logarithmic function with base 2. Its graph passes through the point (1, 0), has a vertical asymptote at x = 0, and is defined for x > 0. The domain of f(x) is (0, ∞), and the range is (-∞, ∞).
2. Step 2: Analyze the transformation in g(x) = log₂(x - 2). The term (x - 2) indicates a horizontal shift of the graph of f(x) to the right by 2 units. This means every point on the graph of f(x) will move 2 units to the right.
3. Step 3: Apply the horizontal shift to the key features of the graph of f(x). The vertical asymptote, originally at x = 0, will now shift to x = 2. The x-intercept, originally at (1, 0), will shift to (3, 0).
4. Step 4: Determine the domain and range of g(x). Since the graph is shifted to the right by 2 units, the domain of g(x) is (2, ∞). The range remains unchanged as (-∞, ∞), because logarithmic functions are not affected vertically by horizontal shifts.
5. Step 5: Summarize the key features of g(x). The x-intercept is at (3, 0), the vertical asymptote is at x = 2, the domain is (2, ∞), and the range is (-∞, ∞). Use these features to sketch the graph of g(x), ensuring it reflects the horizontal shift of the base function.

Key Concepts

Logarithmic Functions

Logarithmic functions, such as f(x) = log2 x, are the inverses of exponential functions. They are defined for positive real numbers and have a characteristic shape that approaches the vertical axis (y-axis) but never touches it, indicating a vertical asymptote at x = 0. Understanding the properties of logarithmic functions is essential for analyzing their graphs and transformations.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For example, g(x) = log2 (x-2) represents a horizontal shift of the graph of f(x) = log2 x to the right by 2 units. Recognizing how these transformations affect the graph is crucial for determining features like intercepts, asymptotes, and the overall shape of the function.

Domain and Range

The domain of a function refers to all possible input values (x-values) for which the function is defined, while the range refers to all possible output values (y-values). For logarithmic functions, the domain is typically restricted to values greater than zero, and the range is all real numbers. Understanding the domain and range helps in identifying the behavior of the function and its graph.