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 properties, including how they behave as x approaches 0 and their growth rate, is essential for graphing and analyzing transformations.
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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. In the case of h(x) = 2 + log₂ x, the '+2' indicates a vertical shift upward by 2 units. Recognizing how these transformations affect the graph helps in determining the new domain, range, and asymptotic behavior of the transformed function.
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Domain & Range of Transformed Functions
Asymptotes and Domain/Range
Asymptotes are lines that a graph approaches but never touches, with vertical asymptotes indicating restrictions on the domain. For the function h(x) = 2 + log₂ x, the vertical asymptote remains at x = 0, while the domain is x > 0. The range, however, shifts due to the vertical transformation, resulting in all real numbers greater than 2.
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Determining Horizontal Asymptotes