Graph the Function f(x) = 5 and Find the Domain and Range

Graph the Function f(x) = 5 and Find the Domain and Range

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[angelmendez-rivera]

This is incorrect. The domain can actually be any set, including the empty set. In particular, the domain can be N, Z, Q, or C. It need not be R. In fact, the equation f(x) = 5 does not specify a function. The domain of a function cannot be uniquely determined by an equation, and neither can be the codomain or the range. To the contrary: for f to be a well-defined function, the domain must be specified in and of itself, and the same is true for the codomain. I could have chosen the domain to be Z, the codomain to be Q, so f : Z —> Q, such that f(x) = 5 everywhere, and this would be a well-defined function. If instead, g : Q —> C, such that g(x) = 5 everywhere, then this still would be a well-defined function, and it is a different function from f. In both cases, the range is {5}, but the domain and codomain are different. In fact, the meaning of the symbol 5 is technically different in both cases. For f, f(x) = 5 is a rational number, whereas g(x) = 5 is a complex number. Despite the fact that we use the same symbol to denote them, these are different, unequal mathematical objects.