The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. g(x) = ln (x+2)

Logarithmic functions, such as f(x) = ln x, are the inverses of exponential functions. The natural logarithm, ln x, is defined for x > 0 and has a vertical asymptote at x = 0. Understanding the properties of logarithmic functions, including their shape and behavior, is essential for analyzing transformations and their effects on the graph.

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For example, g(x) = ln(x + 2) represents a horizontal shift of the graph of f(x) = ln x to the left by 2 units. Recognizing how these transformations affect the graph's position and shape is crucial for accurately graphing the new function.

Asymptotes are lines that a graph approaches but never touches. For the function g(x) = ln(x + 2), the vertical asymptote occurs at x = -2, where the function is undefined. Identifying asymptotes helps determine the behavior of the function near these critical points, which is important for understanding the domain and range of the function.