# Can the image and preimage of a function be the empty set? [closed]

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, user284331Jan 28 '18 at 8:59

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• The only function with empty domain and codomain is the empty function. That is, a function is defined as a set of ordered pairs, no two of which have the same first element. The empty set satisfies this condition, and in this context, it's called "the empty function." – saulspatz Jan 27 '18 at 5:46

In particular $f(1)$ is not well-defined as the rule maps outside the codomain. It does not have exactly one output.