Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Functions
Logarithmic functions, such as ƒ(x) = log₁/₂(x-2), are the inverses of exponential functions. They are defined for positive arguments, meaning the input (x-2) must be greater than zero. This characteristic shapes the function's domain and range, as logarithms can only take positive values.
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Domain of a Function
The domain of a function refers to the set of all possible input values (x) for which the function is defined. For the function ƒ(x) = log₁/₂(x-2), the domain is determined by the condition x-2 > 0, leading to x > 2. Thus, the domain is all real numbers greater than 2.
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Range of a Function
The range of a function is the set of all possible output values (ƒ(x)) that the function can produce. For logarithmic functions, the range is typically all real numbers, as they can extend infinitely in the negative direction. Therefore, for ƒ(x) = log₁/₂(x-2), the range is (-∞, ∞).
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