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)

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1. The base function given is \( f(x) = \ln x \). This is the natural logarithm function, which has a vertical asymptote at \( x = 0 \), a domain of \( (0, \infty) \), and a range of \( (-\infty, \infty) \).
2. The function \( g(x) = \ln(x+2) \) is a horizontal shift of the base function \( f(x) = \ln x \). Specifically, it is shifted 2 units to the left.
3. To graph \( g(x) = \ln(x+2) \), take the graph of \( f(x) = \ln x \) and shift every point 2 units to the left. The vertical asymptote, originally at \( x = 0 \), will now be at \( x = -2 \).
4. The domain of \( g(x) = \ln(x+2) \) is \( (-2, \infty) \) because the function is defined for all \( x \) such that \( x+2 > 0 \). The range remains \( (-\infty, \infty) \) as the logarithmic function can take any real value.
5. The equation of the vertical asymptote for \( g(x) = \ln(x+2) \) is \( x = -2 \). This is where the function approaches negative infinity as \( x \) approaches \(-2\) from the right.

Key Concepts

Logarithmic Functions

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

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

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.