Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Graphs of Logarithmic Functions
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.
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Domain & Range of Transformed Functions
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.
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Introduction to Asymptotes