For example, if I define a function $f(x) = -x$ with a domain and codomain of positive numbers (in other words a nonexistant image and preimage) would this still be a function?
closed as unclear what you're asking by Rohan, Lord Shark the Unknown, Cameron Williams, Namaste, user284331 Jan 28 '18 at 8:59
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No, it is not a function.
In particular $f(1)$ is not well-defined as the rule maps outside the codomain. It does not have exactly one output.