Here are the essential concepts you must grasp in order to answer the question correctly.
Function Composition
Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, (g∘ƒ)(x) means applying function f first and then applying function g to the result. Understanding how to correctly perform this operation is crucial for analyzing the behavior of the composed function.
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Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the functions f(x) = 1/(x-2) and g(x) = 1/x, it is essential to identify values that would make the denominator zero, as these values are excluded from the domain. This understanding is vital for determining the domain of the composed function (g∘ƒ)(x).
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Domain Restrictions of Composed Functions
Vertical Asymptotes
Vertical asymptotes occur in rational functions where the function approaches infinity as the input approaches a certain value. For the functions given, vertical asymptotes can be found at the values that make the denominator zero. Recognizing these asymptotes helps in understanding the behavior of the functions and their compositions, particularly in relation to their domains.
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Determining Vertical Asymptotes