Here are the essential concepts you must grasp in order to answer the question correctly.
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 vertical asymptote at x = 0. Understanding their basic shape and properties, including how they increase and their domain and range, is essential for graphing and transforming these functions.
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Graphs of Logarithmic Functions
Transformations of Functions
Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For example, the function g(x) = (1/2)log₂ x represents a vertical compression of f(x) = log₂ x by a factor of 1/2. Recognizing how these transformations affect the graph helps in accurately sketching the new function and understanding its characteristics.
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Domain & Range of Transformed Functions
Asymptotes, Domain, and Range
Asymptotes are lines that a graph approaches but never touches, with vertical asymptotes indicating values where the function is undefined. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For g(x) = (1/2)log₂ x, identifying the vertical asymptote, domain, and range is crucial for understanding the behavior of the function.
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Determining Horizontal Asymptotes