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: Graph the basic function f(x) = \log_2{x}.
2. Step 2: Identify the key features of f(x) = \log_2{x}, such as the x-intercept at (1,0) and the vertical asymptote at x=0.
3. Step 3: Apply the transformation to f(x) to obtain g(x) = \log_2{(x-2)}. This involves a horizontal shift to the right by 2 units.
4. Step 4: Determine the new x-intercept and vertical asymptote for g(x). The x-intercept is now at (3,0) and the vertical asymptote is at x=2.
5. Step 5: Use the transformed graph to determine the domain and range of g(x). The domain is (2, \infty) and the range is (-\infty, \infty).

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 basic properties of logarithms, including their domain and range, is essential for analyzing transformations.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For g(x) = log2 (x-2), the graph of f(x) = log2 x is shifted to the right by 2 units. This transformation affects the function's x-intercept and vertical asymptote, which are crucial for determining the graph's behavior and characteristics.

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 g(x) = log2 (x-2), the domain is x > 2, as the logarithm is only defined for positive arguments. The range remains all real numbers, reflecting the behavior of logarithmic functions, which can produce any real output.