The figure shows the graph of f(x) = log x. In Exercises 59–64, 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) = 1-log x

Verified step by step guidance

1. Start with the graph of f(x) = \log x, which has a vertical asymptote at x = 0.
2. The function g(x) = 1 - \log x is a vertical reflection of f(x) = \log x across the x-axis, followed by a vertical shift upwards by 1 unit.
3. Reflect the graph of f(x) = \log x across the x-axis to get -\log x.
4. Shift the graph of -\log x upwards by 1 unit to obtain g(x) = 1 - \log x.
5. The domain of g(x) is (0, \infty) and the range is (-\infty, 1). The vertical asymptote remains at x = 0.

Key Concepts

Logarithmic Functions

Logarithmic functions, such as f(x) = log x, are the inverses of exponential functions. They are defined for positive real numbers and have a vertical asymptote at x = 0. The graph of log x increases slowly and passes through the point (1, 0), indicating that log(1) = 0. Understanding the properties of logarithmic functions is essential for analyzing transformations and their effects on the graph.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For example, the function g(x) = 1 - log x reflects the graph of f(x) = log x across the x-axis and shifts it upward by 1 unit. Recognizing how these transformations affect the graph's shape and position is crucial for accurately graphing new functions based on existing ones.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For logarithmic functions, there is a vertical asymptote at x = 0, indicating that the function is undefined for non-positive values of x. When transforming functions, it is important to determine how the asymptotes change, as they provide critical information about the function's behavior and help define its domain and range.