From http://semver.org:
Given a version number MAJOR.MINOR.PATCH, increment the:
MAJOR version when you make incompatible API changes,
MINOR version when you add functionality in a backwards-compatible manner, and
PATCH version when you make backwards-compatible bug fixes.
So, in your case, if you don't also maintain the old function name, in order to retain compatibility with older versions of the API you should increment the major version number.
One way to look at it, in order to know if compatibility is broken, would be to imagine that your API and functionality is encapsulated in a library which offers this functionality to other programs. You now make changes to that API. If the programs which linked to the old version of your API need to be changed in order to use the new version of your library, you have broken compatibility and the major version should be changed. You may solve this problem by overriding and maintaining the deprecated old function calls, but it would increase the complexity of the API.